Tautology

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					Mathematical Logic                                                                 U = the set of all human beings
                                                                                   S = the set of all scholars
Introduction                                                                  And T = the set of all teachers
     Logic was extensively developed in Greece. In the middle                 Clearly, S ⊂ U and T ⊂ U
ages the treatises of Aristotle concerning logic were re-                     According to the above statement , if follows that T ⊂ S .
discovered. The axiomatic approach to logic was first proposed            Thus, the truth of the above statement can be represented by the
by George Boole. On this account logic relative to mathematics            Venn diagram shown in
is sometimes called Boolean logic. It is also called mathematical                                                       U
logic or more recently symbolic logic.                                                                             S
                                                                                                                   T
     The dictionary meaning of the word ‘Logic’ is “the science of
reasoning”. It is the study and analysis of the nature of valid                                  x
arguments. In the process of reasoning we communicate our
ideas or thoughts with the help of sentences in a particular
language. The following types of sentences are normally used in                Now, if we consider the statement : “ There are some
our every day communication.                                              scholars who are teachers”, It is evident from the Venn diagram
     (1) Assertive sentence           (2) Imperative sentence             that there is a scholar x who is not a teacher. Therefore, the
                                                                          above statement is false and its truth value is ‘F’. Thus, we can
     (3) Exclamatory sentence         (4) Interrogative sentence          also check the truth and falsity of other statements which are
     In this chapter, we shall be discussing about a specific type        connected to a given statement.
of sentences which will called as statement or propositions.
                                                                          Types of statements
Statements or propositions                                                     In Mathematical logic, we generally come across two types
      Propositions : A statement or a proposition is an                   of statements or proposition, namely, simple statements and
assertive (or declarative) sentence which is either true or false         compound statements as defined below.
but not both a true statement is called valid statement. If a                  (i) Simple statements : Any statement or proposition
statement is false, then it is called invalid statement.                  whose truth value does not explicity depend on another
                                                                          statement is said to be a simple statement.
      Open statement : A declarative sentence containing
variable (s) is an open statement if it becomes a statement when               In other words, a statement is said to be simple if it cannot
the variable (s) is (are) replaced by some definite value (s).            be broken down into simpler statements, that is, if it is not
                                                                          composed of simpler statements.
      Truth Set : The set of all those values of the variable (s) in
                                                                               (ii) Compound statements : If a statement is
an open statement for which it becomes a true statement is
                                                                          combination of two or more simple statements, then it is said to
called the truth set of the open statement.
                                                                          be a compound statement or a compound proposition.
      Truth Value : The truth or falsity of a statement is called
its truth value.                                                          Truth tables
      If a statement is true, then we say that its truth value is              Definition : A table that shows the relationship between
‘True’ or ‘T’. On the other hand the truth value of a false               the truth value of a compound statement S (p, q, r,..) and the
statement is ‘False’ or ‘F’.                                              truth values of its sub-statement p, q, r, .....etc, is called the
      Logical variables : In the study of logic, statements are           truth table of statement S.
represented by lower case letters such as p, q, r, s.. These letters           Construction of truth table : In order to construct the
are called logical variables.                                             truth table for a compound statement, we first prepare a table
      For example, the statement ‘The sun is a star’ may be               consisting of rows and columns. At the top of the initial
represented or denoted by p and we write                                  columns, we write the variables denoting the sub-statements or
                      p : The sun is a star                               constituent statements and then we write their truth values, in
      Similarly, we may denote the statement                              the last column. We write the truth value of the compound
                      14 – 5 =– 2.                                        statement on the basis of the truth values of the constituent
      Quantifiers : The symbol ∀ (stands for ‘for all’) and               statements written in the initial columns. If a compound
 ∃ (stands for “there exists”) are known as quantifiers.                  statement is made up of two simple statement, then the number
      In other word, quantifiers are symbols used to denote a             of rows in the truth table will be 2 2 and if it is made up of three
group of words or a phrase.
                                                                          simple statements, then the number of rows will be 2 3 . In
      The symbols ∀ and ∃ are known as existential quantifiers.
                                                                          general, if the compound statement is made up of n sub-
An open sentence used with quantifiers always becomes a
statement.                                                                statements, then its truth table will contain 2 n rows.
      Quantified statements : The statements containing                   Basic logical connectives or logical operators
quantifiers are known as quantified statements.
       x 2 > 0 .∀x ∈ R is a quantified statement. Its truth value is T.        Definition : The phrases or words which connect simple
                                                                          statements are called logical connectives or sentential
Use of venn diagrams in checking truth and                                connectives or simply connectives or logical operators.
falsity of statements                                                          In the following table, we list some possible connectives,
     In this section, we shall discuss how Venn diagrams are              their symbols and the nature of the compound statement
used to represent truth and falsity of statements or propositions.        formed by them.
For this, let us consider the statement: “All teachers are                      Connective Symbol        Nature of the compound
scholars”. Let us assume that this statement is true. To                                                 statement formed by using the
represent the truth of the above statement, we define the                                                connective
following sets                                                                  and              ∧       Conjunction




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      or                     ∨         disjunction                                     Logically equivalent statement : Two compound
      If....then          ⇒ or →       Implication or conditional               S 1 ( p, q, r,...) and S 2 ( p, q, r...) are said to be logically equivalent,
      If and only         ⇔ or ↔       Equivalence or bi-conditional           or simply equivalent if they have the same truth values for all
      if (iff)                                                                 logically possibilities.
      not                         Negation                                             If statements S 1 ( p, q, r,...) and S 2 ( p, q, r...) are logically
                     ~ or ┓
     (i) Conjunction : Any two simple statements can be                        equivalent, then we write
connected by the word “and” to form a compound statement                                S 1 ( p, q, r,...) ≡ S 2 ( p, q, r...)
called the conjunction of the original statements.                                  It follows from the above definition that two statements S 1
     Symbolically if p and q are two simple statements, then
 p ∧ q denotes the conjunction of p and q and is read as “p and                and S 2 are logically equivalent if they have identical truth tables
q”.                                                                            i.e., the entries in the last column of the truth tables are same.
     (ii) Disjunction or alternation Any two statements can                    Negation of compound statements
be connected by the word “or” to form a compound statement
                                                                                    We have learnt about negation of a simple statement.
called the disjunction of the original statements.
                                                                               Writing the negation of compound statements having
     Symbolically, if p and q are two simple statements, then                  conjunction, disjunctions, implication, equivalence, etc, is not
 p ∨ q denotes the disjunction of p and q and is read as “ p or q”.            very simple. So, let us discuss the negation of compound
     (iii) Negation : The denial of a statement p is called its                statement.
negation, written as ~ p.                                                           (i) Negation of conjuntion :
     Negation of any statement p is formed by writing “ It is not                   If p and q are two statements, then ~ ( p ∧ q ) ≡ (~ p ∨ ~ q )
the case that ..... “ or “ It is false that.......” before p or, if possible
by inserting in p the word “not”.                                                   (ii) Negation of disjuntion :
     • Negation is called a connective although it does not                             If    p    and    q are   two               statements,        then
combine two or more statements. In fact, it only modifies a                    ~ ( p ∨ q ) ≡ (~ p ∧ ~ q )
statement.                                                                          (iii) Negation of implication :
     (iv) Implication or conditional statements : Any two                           If p and q are two statements, then ~ ( p ⇒ q ) = ( p ∧ ~ q )
statements connected by the connective phrase “if.. then” give
rise to a compound statement which is known as an implication                     (iv) Negation of biconditional                        statement        or
or a conditional statement.                                                    equivalence :
     If p and q are two statements forming the implication ‘if p                  If p and q are two statements, then
then q′, then we denote this implication by                                        ~ ( p ⇔ q ) = ( p ∧ ~ q ) ∨ (q ∧ ~ p )
      " p ⇒ q " or " p → q " .
                                                                               Tautologies and contradictions
    In the implication " p ⇒ q " , p is the antecedent and q is the
                                                                                     Let p, q, r,.... be statements, then any statement involving
consequent.
                                                                                p, q, r ,....and the logical connectives ∧,∨, ~, ⇒, ⇔ is called a
    Truth table for a conditional a statement
                                                                               statement pattern or a Well Formed Formula (WFF).
             p                      q                p⇒q
                                                                                     For example
                 T                      T                      T                     (i) p ∨ q
                 T                      F                      F
                                                                                    (ii)   p⇒q
                 F                      T                      T
                 F                      F                      T                    (iii) (( p ∧ q) ∨ r) ⇒ (s ∧ ~ s)
                                                                                    (iv) ( p ⇒ q) ⇔ (~ q ⇒~ p) etc.
     (v) Biconditional statement : A statement is a
biconditional statement if it is the conjunction of two                              are statement patterns.
conditional statements (implications) one converse to the other.                     A statement is also a statement pattern.
     Thus, if p and q are two statements, then the compound                          Thus, we can define statement pattern as follows.
statement      p ⇒ q and q ⇒ p is called a biconditional                             Statement pattern : A compound statement with the
statements or an equivalence and is denoted by p ⇔ q .                         repetitive use of the logical connectives is called a statement
                                                                               pattern or a well- formed formula.
     Thus, p ⇔ q : ( p ⇒ q ) ∧ (q ⇒ p )
                                                                                     Tautology : A statement pattern is called a tautology, if it
     Truth table for a biconditional statement : Since                         is always true, whatever may be the truth values of constitute
 p ⇔ q is the conjunction of p ⇒ q and q ⇒ p . So, we have                     statements.
the following truth table for p ⇔ q .                                                A tautology is called a theorem or a logically valid statement
       p     q       p⇒q       q⇒p          p ⇔ q = ( p ⇒ q ) ∧ (q ⇒ p )       pattern. A tautology, contains only T in the last column of its
                                                                               truth table.
       T     T        T            T                     T
       T     F        F            T                     F
                                                                                     Contradiction : A statement pattern is called a
                                                                               contradiction, if it is always false, whatever may the truth values
       F     T        T            F                     F
                                                                               of its constitute statements.
       F     F        T            T                     T
                                                                                     In the last column of the truth table of contradiction there is
Logical equivalence                                                            always F.
                                                                                     • The negation of a tautology is a contradiction and vice
                                                                               versa.




2|Pa g e
Algebra of statements
     In the previous section, we have seen that statements
satisfy many standard results. In this section, we shall state                         Mathematical logic(QUESTIONS)
those results as laws of algebra of statements.
     The following are some laws of algebra of statements.                     1.    Which of the following is a statement
     (i) Idempotent laws : For any statement p, we have                              (a) Open the door              (b) Do your homework
     (a) p ∨ p ≡ p                     (b) p ∧ p ≡ p                                 (c) Switch on the fan          (d) Two plus two is four
                                                                               2.    Which of the following is a statement
    (ii) Commutative laws : For any two statements p and q,
                                                                                     (a) May you live long !
we have
                                                                                     (b) May God bless you !
    (a) p ∨ q ≡ q ∨ p           (b) p ∧ p ≡ q ∧ p
                                                                                     (c) The sun is a star
    (iii) Association laws : For any three statements p, q, r,                       (d) Hurrah ! we have won the match
we have                                                                        3.    Which of the following is not a statement
    (a) ( p ∨ q) ∨ r ≡ p ∨ (q ∨ r) (b) ( p ∧ q) ∧ r ≡ p ∧ (q ∧ r)                    (a) Roses are red
    (iv) Distributive laws : For any three statements p, q, r                        (b) New Delhi is in India
we have                                                                              (c) Every square is a rectangle
    (a) p ∧ ( p ∨ q) ≡ ( p ∧ q) ∨ (q ∧ r)                                            (d) Alas ! I have failed
                                                                               4.    Which of the following is not a statement
     (b) p ∨ ( p ∧ q) ≡ ( p ∨ q) ∧ (q ∨ r)                                           (a) Every set is a finite set
     (v) Demorgan’s laws : If p and q are two statements, then                       (b) 8 is less than 6
     (a) ~ ( p ∧ q) ≡~ p ∨ ~ q    (b) ~ ( p ∨ q) ≡~ p ∧ ~ q                          (c) Where are you going ?
    (vi) Identity laws : If t and c denote a tautology and a                         (d) The sum of interior angles of a triangle is 180 degrees
contradiction respectively, then for any statement p, we have                  5.    Which of the following is not a statement
    (a) p ∧ t ≡ p (b) p ∨ c ≡ p (c) p ∨ t ≡ t (d) p ∧ c ≡ c                          (a) Please do me a favour (b) 2 is an even integer
                                                                                     (c) 2 + 1 = 3                  (d) The number 17 is prime
     (vii) Complement laws : For any statements p, we have
                                                                               6.    Which of the following is not a statement
     (a) p ∨ ~ p = t (b) p ∧ ~ p = c (c) ~ t = c (d) ~ c = t                         (a) Give me a glass of water
    where t and c denote a tautology and a contradiction                             (b) Asia is a continent
respectively.                                                                        (c) The earth revolved round the sun
    (viii) Law of contrapositive : For any two statements p                          (d) The number 6 has two prime factors 2, 3
and q, we have                                                                 7.    Which of the following is an open statement
     p ⇒ q ≡~ q ⇒~ p                                                                 (a) x is a natural number (b) Give me a glass of water
    (ix) Involution laws : For any statement p, we have                              (c) Wish you best of luck (d) Good morning to all
~ (~ p ) ≡ p                                                                   8.    Negation of the conditional : “If it rains, I shall go to
                                                                                     school” is
Duality                                                                              (a) It rains and I shall go to school
                                                                                     (b) It rains and I shall not go to school
     Definition : Two compound statements S 1 and S 2 are
                                                                                     (c) It does not rains and I shall go to school
said to be duals of each other if one can be obtained from the                       (d) None of these
other by replacing ∧ by ∨ and ∨ by ∧ .
                                                                               9.    Negation of “Paris in France and London is in England” is
     • The connective ∧ and ∨ are also called duals of each                          (a) Paris is in England and London is in France
other                                                                                (b) Paris is not in France or London is not in England
     • If a compound statements contains the special variable                        (c) Paris is in England or London is in France
t (tautology) or c (contradiction), then to obtain its dual we                       (d) None of these
replace t by c and c by t in addition to replacing ∧ by ∨ and                  10.   Negation is “2 + 3 = 5 and 8 < 10” is
 ∨ by ∧ .
                                                                                     (a) 2 + 3 ≠ 5 and < 10         (b) 2 + 3 = 5 and 8 ≮ 10
     • Let S ( p, q ) be a compound statement containing two sub-
                                                                                     (c) 2 + 3 ≠ 5 or 8 ≮ 10        (d) None of these
statements and S * (p, q) be its dual. Then,
                                                                               11.   Negation of “Ram is in Class X or Rashmi is in Class XII”
     (i) ~ S ( p, q ) ≡ S * (~ p, ~ q )                                              is
                                                                                     (a) Ram is not in class X but Ram is in class XII
     (ii) ~ S * ( p, q ) ≡ S (~ p, ~ q )
                                                                                     (b) Ram is not in class X but Rashmi is not in class XII
      • The above result can be extended to the compound                             (c) Either Ram is not in class X or Ram is not in class XII
statements having finite number of sub- statements. Thus, if                         (d) None of these
S ( p1 , p 2 ,.... pn ) is a compound statement containing n sub-
                                                                               12.   The conditional ( p ∧ q ) ⇒ p is
statement p1 , p 2 ,...., pn and S * ( p1 p 2 ,...., pn ) is its dual. Then,         (a)   A tautology
     (i) ~ S ( p1 , p 2 ,...., pn ) ≡ S * (~ p1 , ~ p 2 ,...., ~ pn )                (b)   A fallacy i.e., contradiction
                                                                                     (c)   Neither tautology nor fallacy
     (ii) ~ S * ( p1 , p 2 ,...., pn ) ≡ S (~ p1 , ~ p 2 ,...., ~ pn )
                                                                                     (d)   None of these




3|Pa g e
13.   Which of the following is a contradiction                               (c) Either (a) or (b)            (d) Neither (a) nor (b)
      (a) ( p ∧ q )∧ ~ ( p ∨ q )    (b) p ∨ (~ p ∧ q )                  27.   Which of the following is not logically equivalent to the
                                                                              proposition : “A real number is either rational or
      (c) ( p ⇒ q ) ⇒ p              (d) None of these                        irrational”.
14.   Which of the following is logically equivalent to                       (a) If a number is neither rational nor irrational then it is
      ~ (~ p ⇒ q )                                                                  not real
                                                                              (b) If a number is not a rational or not an irrational, then
      (a) p ∧ q                      (b) p ∧ ~ q                                    it is not real
      (c) ~ p ∧ q                    (d) ~ p ∧ ~ q                            (c) If a number is not real, then it is neither rational nor
                                                                                    irrational
15.   ~ ( p ∨ q) is equal to                                                  (d) If a number is real, then it is rational or irrational
      (a) ~ p ∨ ~ q                  (b) ~ p ∧ ~ q                      28.   If p : It rains today, q : I go to school, r : I shall meet any
                                                                              friends and s : I shall go for a movie, then which of the
      (c) ~ p ∨ q                    (d) p ∨ ~ q                              following is the proposition :
16.   ~ ( p ∧ q) is equal to                                                  If it does not rain or if I do not go to school, then I shall
                                                                              meet my friend and go for a movie.
      (a) ~ p ∨ ~ q                  (b) ~ p ∧ ~ q                            (a) ~ ( p ∧ q ) ⇒ (r ∧ s)        (b) ~ ( p ∧ ~ q) ⇒ (r ∧ s)
      (c) ~ p ∧ q                    (d) p ∧ ~ q                              (c) ~ ( p ∧ q ) ⇒ (r ∨ s)      (d) None of these
17.   (~ (~ p)) ∧ q is equal to                                         29.   The negation of the compound proposition p ∨ (~ p ∨ q )
      (a) ~ p ∧ q                    (b) p ∧ q                                is
                                                                              (a) ( p ∧ ~ q ) ∧ ~ p     (b) ( p ∧ ~ q ) ∨ ~ p
      (c) p ∧ ~ q                    (d) ~ p ∧ ~ q
                                                                              (c) ( p ∨ ~ q ) ∨ ~ p         (d) None of these
18.   ~ ( p ∨ (~ q)) is equal to
                                                                        30.   Which of the following is true
      (a) ~ p ∨ q                    (b) (~ p) ∧ q                            (a) p ⇒ q ≡ ~ p ⇒ ~ q
      (c) ~ p ∨ ~ p                  (d) ~ p ∧ ~ q                            (b) ~ ( p ⇒ ~ q ) ≡ ~ p ∧ q
19.   ~ ((~ p ) ∧ q ) is equal to                                             (c) ~ (~ p ⇒ ~ q ) ≡~ p ∧ q
                                                                              (d) ~ ( p ⇔ q ) ≡ [~ ( p ⇒ q )∧ ~ (q ⇒ p )]
      (a) p ∨ (~ q)                  (b) p ∨ q
                                                                        31.   ~ ( p ∨ q ) ∨ (~ p ∧ q ) is logically equivalent to
      (c) p ∧ (~ q)                  (d) ~ p ∧ ~ q
                                                                              (a) ~p                            (b) p
20.   ~ ( p ⇔ q) is                                                           (c) q                             (d) ~q
                                                                        32.   The inverse of the proposition ( p ∧ ~ q ) ⇒ r is
      (a) ~ p ∧ ~ q                  (b) ~ p ∨ ~ q
                                                                              (a) ~ r ⇒ ~ p ∨ q                 (b) ~ p ∨ q ⇒ ~ r
      (c) ( p ∧ ~ q ) ∨ (~ p ∧ q )   (d) None of these
                                                                              (c) r ⇒ p ∧ ~ q                   (d) None of these
21.   p ⇒ q can also be written as                                      33.   When does the current flow through the following circuit
      (a) p ⇒ ~ q                    (b) ~ p ∨ q                              (a) p, q, r should be closed          q
      (c) ~ q ⇒~ p                   (d) None of these                        (b) p, q, r should be open
                                                                              (c) Always                  p                        r
22.   If p, q, r are simple propositions with truth values T, F, T,           (d) None of these
      then the truth value of (~ p ∨ q ) ∧ ~ r ⇒ p is                                                               q
                                                                        34.   Which Venn diagram represent the truth of the statement
      (a) True                        (b) False                               “All students are hard working.”
      (c) True if r is false          (d) True if q is true                   Where U = Universal set of human beings
23.   If ( p ∧ ~ r) ⇒ (q ∨ r) is false and q and r are both false,                    S = Set of all students
                                                                                      H = Set of all hard workers
      then p is
                                                                                                      U                        S    H   U
      (a) True                        (b) False                                             H
      (c) May be true or false        (d) Data insufficient                   (a)           S
                                                                                                             (b)
24.   If p, q, r are simple propositions, then ( p ∧ q ) ∧ (q ∧ r) is
      true then
      (a) p, q, r are all false                                                                       U
      (b) p, q, r are all true                                                              S=H
      (c) p, q are true and r is false                                        (c)                         (d) None of these
                                                                        35.   Which Venn diagram represent the truth of the
      (d) p is true and q and r are false
                                                                              statements “No child is naughty”
25.   ~ ( p ⇒ q ) ⇔~ p ∨ ~ q is
                                                                              Where U = Universal set of human beings
      (a) A tautology
                                                                                    C = Set of children
      (b) A contradiction
      (c) Neither a tautology nor a contradiction                                   N = Set of naughty persons
      (d) Cannot come to any conclusion                                                 N         C
26.   ( p ∧ ~ q ) ∧ (~ p ∨ q ) is                                                                                          N
                                                                              (a)                            (b)           C
      (a) A contradiction             (b) A tautology
                                                                                    U                              U




4|P a g e
                                                                      42.   The contrapositive of ( p ∨ q ) ⇒ r is[Karnataka CET 1999]
                                                                            (a) r ⇒ ( p ∨ q )                    (b) ~ r ⇒ ( p ∨ q )
                    N       C
                                                                            (c) ~ r ⇒ ~ p ∧ ~ q                  (d) p ⇒ (q ∨ r)
      (c)                            (d) None of these
                                                                      43.   If p ⇒ (q ∨ r) is false, then the truth values of p, q, r are
            U                                                               respectively                         [Karnataka CET 2000]
36.   Which Venn diagram represent the truth of the statement               (a) T, F, F                    (b) F, F, F
      “No policeman is a thief”                                             (c) F, T, T                    (d) T, T, F
                                                                      44.   The logically equivalent proposition of p ⇔ q is
                                                                                                                       [Karnataka CET 2000]
                P               T                 P           T             (a) ( p ∧ q ) ∨ ( p ∧ q )            (b) ( p ⇒ q ) ∧ (q ⇒ p )
      (a)                            (b)
            U                              U                                (c) ( p ∧ q ) ∨ (q ⇒ p )             (d) ( p ∧ q ) ⇒ (q ∨ p )
                                                                      45.   The false statement in the following is
                                                                                                                          [Karnataka CET 2002]
                        P
      (c)               T            (d) None of these                      (a) p ∧ (~ p ) is a contradiction
                                                                            (b) ( p ⇒ q ) ⇔ (~ q ⇒ ~ p ) is a contradiction
            U
37.   Which Venn diagram represent the truth of the statement               (c) ~ (~ p ) ⇔ p is a tautology
      “Some teenagers are not dreamers”                                     (d) p ∨ (~ p) ⇔ is a tautology

                                                  T       D           46.   If p ⇒ (~ p ∨ q ) is false, the truth values of p and q are
                                                                            respectively                         [Karnataka CET 2002]
      (a)       T               D    (b)                                    (a) F, T                       (b) F, F
            U
                                                                            (c) T, T                       (d) T, F
                                           U
                                                                      47.   Which of the following is not a proposition
                    T       D                                                                                             [Karnataka CET 2002]
      (c)                            (d) None of these                      (a)     3 is a prime

            U                                                               (b)    2 is irrational
                                                                            (c) Mathematics is interesting
38.   Which of the following Venn diagram corresponds to the
      statement                                                             (d) 5 is an even integer
      “All mothers are women”                                         48.   ( p ∧ ~ q ) ∧ (~ p ∧ q ) is                   [Karnataka CET 2003]
      (M is the set of all mothers, W is the set of all women)              (a) A tautology
                                                  W       M
                                                                            (b) A contradiction
      (a)                            (b)                                    (c) Both a tautology and a contradiction
                W               M                                           (d) Neither a tautology nor a contradiction
                                                                      49.   ~ p ∧ q is logically equivalent to [Karnataka CET 2004]
            U                              U
                                                                            (a) p → q                            (b) q → p
                        W                             M                     (c) ~ ( p → q)                       (d) ~ (q → p )
      (c)               M            (d)              W
                                                                      50.   Which of the following is the inverse of the proposition :
                                                                            “If a number is a prime then it is odd.”[Karnataka CET 2004]
            U                              U
                                                                            (a) If a number is not a prime then it is odd
39.   The negative of q ∨ ~ ( p ∧ r) is        [Karnataka CET 1997]         (b) If a number is not a prime then it is odd
      (a) ~ q ∧ ~ ( p ∧ r)           (b) ~ q ∧ ( p ∧ r)                     (c) If a number is not odd then it is not a prime
                                                                            (d) If a number is not odd then it is a prime
      (c) ~ q ∨ ( p ∧ r)             (d) None of these
40.   The propositions ( p ⇒ ~ p ) ∧ (~ p ⇒ p ) is a                                              ANSWERS
                                               [Karnataka CET 1997]
      (a) Tautology and contradiction                                                        Mathematical logic
      (b) Neither tautology nor contradiction
      (c) Contradiction                                                1     d        2       c         3    d        4       c        5    a
      (d) Tautology                                                    6     a        7       a         8    b        9       b        10   c
41.   Which of the following is always true[Karnataka CET 1998]
                                                                       11    d        12      a         13   a        14      d        15   b
      (a) ( p ⇒ q ) ≡ ~ q ⇒ ~ p    (b) ~ ( p ∨ q ) ≡ ∨ p ∨ ~ q
      (c) ~ ( p ⇒ q ) ≡ p ∧ ~ q      (d) ~ ( p ∨ q ) ≡ ~ p ∧ ~ q       16    a        17      b         18   b        19      a        20   c




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 21    b       22      a      23     a        24      b     25       c               T   T      T         T          F                F
                                                                                     T   F      F         T          F                F
 26    a       27      b      28     a        29      a     30       c
                                                                                     F   T      F         T          F                F
 31    a       32      b      33     a        34      a     35       a               F   F      F         F          T                F
 36    a       37      c      38     c        39      b     40       c              ∴ ( p ∧ q ) ∧ (~ ( p ∨ q )) is a contradiction.
 41    c       42      c      43     a        44      b     45       b   14.   (d) Since ~ ( p ⇒ q ) ≡ p ∧ ~ q

 46    d       47      c      48     b        49      d     50       b              ~ (~ p ⇒ q ) = ~ p ∧ ~ q
                                                                         15.   (b) ~ ( p ∨ q ) ≡ ~ p ∧ ~ q .
                                                                         16.   (a) ~ ( p ∧ q ) ≡ ~ p ∨ ~ q .
                                                                         17.   (b) (~ (~ p )) ∧ q = p ∧ q .
                                                                         18.   (b) ~ ( p ∨ (~ q)) ≡ ~ p ∧ ~ (~ q ) ≡ (~ p ) ∧ q .
                                                                         19.   (a) ~ ((~ p ) ∧ q ) ≡ ~ (~ p ) ∨ ~ q ≡ p ∨ (~ q ) .
       Mathematical logic(SOLUTIONS)
                                                                         20.   (c) ~ ( p ⇔ q ) = ( p ∧ ~ q ) ∨ (q ∧ ~ p ) .
1.    (d)   “Two plus two is four” is a statement.                       21.   (b) p ⇒ q ≡ ~ p ∨ q .
2.    (c)   “The sun is a star” is a statement.                          22.   (a) ~ p ∨ q means F ∨ F = F, ~r means F (~ p ∨ q )∧ ~ r
3.    (d)   “Alas ! I have failed” is not a statement.                             means F
4.    (c)   “Where are you going?” is not a statement.                             ∵ [(~ p ∨ q ) ∧ ~ r] ⇒ p means T
5.    (a)   “Please do me a favour” is not a statement.
                                                                                                                  [ ∵ in p ⇒ q we have FTT]
6.    (a)   “Give me a glass of water” is not a statement.
                                                                         23.   (a) Given result means p ∧ ~ r is true, q ∨ r is false.
7.    (a)   “x is a rational number” is an open statement.
8.    (b)   p : It rains, q : I shall go to school                       24.   (b) ( p ∧ q ) ∧ (q ∧ r) is true means p ∧ q , q ∧ r are both
            Thus, we have p ⇒ q                                                    true. ⇒ p, q, r are all true.
                                                                         25.   (c)
            Its negation is ~ ( p ⇒ q) i.e. p ∧ ~ q
                                                                                    p q p⇒q ~(p⇒q) ~p ~q ~p∨~q ~(p⇒q)
          i.e. It rains and I shall not go to school.                                                                              ⇔
9.    (b) Let p : Paris is in France, q : London is in England                                                                     ~p∨~q
            ∴ we have p ∧ q                                                         T T T                F     F   F        F         T
            Its negation is ~ ( p ∧ q ) = ~ p ∨ ~ q                                 T F        F         T     F   T        T         T
                                                                                    F T T                F     T   F        T         F
         i.e. Paris is not in France or London is not in
England.                                                                            F F        T         F     T   T        T         F
10. (c) Let p : 2+ 3 = 5, q : 8 < 10                                               Last column shows that result is neither a tautology
                                                                                   nor a contradiction.
         Given proposition is : p ∧ q
                                                                         26.   (a)
            Its negation is ~ ( p ∧ q ) = ~ p ∨ ~ q
                                                                                     p q     ~p      ~q       p∧~q       ~p∨q   (p∧~q)∧(~p∨q)
          ∴ we have 2+ 3 ≠ 5 or 8 ≮ 10.                                              T   T    F       F         F         T           F
11.   (d) Let p : Ram is in Class X, q : Rahim is in class XII                       T   F    F       T        T          F           F
          Given proposition is p ∨ q                                                 F   T    T       F         F         T           F
            Its negation is ~ ( p ∨ q ) = ~ p ∧ ~ q                                  F   F    T       T         F         T           F
                                                                                    Clearly, ( p ∧ ~ q ) ∧ ( p ∨ ~ q ) is a contradiction.
            i.e. Ram is not in class X and Rahim is not in class
            XII.                                                         27.   (b) It is correct.
12.   (a)
                                                                                    ∵ 3 is not rational but it is real.
                 p           q           p∧q              ( p ∧ q) ⇒ p
                 T          T              T                     T       28.   (a) Correct result is (~ p ∨ ~ q ) ⇒ (r ∧ s)
                 T          F              F                     T
                                                                                    So, ~ ( p ∧ q ) ⇒ (r ∧ s) .
                 F          T              F                     T
                 F          F              F                     T       29.   (a) ~ [ p ∨ (~ p ∨ q )] ≡ ~ p ∧ ~ (~ p ∨ q )
            ∴ ( p ∧ q ) ⇒ p is a tautology.                                         ≡ ~ p ∧ (~ (~ p ) ∧ ~ q )
13.   (a)
                                                                                    ≡~ p ∧ ( p ∧ ~ q ) .
             p q     p∧ q    p∨ q    ~ ( p ∨ q)    (p ∧ q) ∧ ~ (p ∨ q)




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30.   (c) ~ ( p ⇒ q ) ≡ p ∧ ~ q                                           46.   (d) p ⇒ (~ p ∨ q ) is false means p is true and ~ p ∨ q is
                                                                                    false.
            ∴ ~ (~ p ⇒ ~ q ) ≡ ~ p ∧ ~ (~ q ) ≡ ~ p ∧ q
                                                                                    ⇒ p is true and both ~p and q are false.
            Thus ~ (~ p ⇒ ~ q ) ≡ ~ p ∧ q .
                                                                                    ⇒ p is true and q is false.
31.   (a) ~ ( p ∨ q ) ∨ (~ p ∨ q ))
                                                                          47.   (c) Mathematics is interesting is not a logical sentence. It
            ≡ (~ p ∧ ~ q ) ∨ (~ p ∧ q )                                             may be interesting for some persons are may not be
                                                                                    interesting for others.
            ≡ ~ p ∧ (~ q ∨ q ) ≡ ~ p .
                                                                                    ∴ This is not a propositions.
32.   (b) Inverse of p ⇒ q is ~ p ⇒ ~ q
                                                                          48.   (b) ( p ∧ ~ q) ∧ (~ p ∧ q) = ( p ∧ ~ p) ∧ (~ q ∧ q) = f ∧ f = f .
            ∴ inverse of ( p ∧ ~ q ) ⇒ r is
                                                                                    (By using associative laws and commutatine laws)
            ~ ( p ∧ ~ q ) ⇒ ~ r i.e. (~ p ∨ q ) ⇒ ~ r .                             ∴ ( p ∧ ~ q ) ∧ (~ p ∧ q ) is a contradiction.
33.   (a) p, q, r should be closed for the current to flow.               49.   (d) ~ p ∧ q =~ (q → p ) .
34.   (a) All students are hard working means S ⊆ H.                      50.   (b) p : A number is a prime.
35.   (a) “No child is naughty” means C ∩ N = φ                                     Q : It is odd.
            i.e. there is no common element between C and N.                        We have p ⇒ q
36.   (a) No policeman is a thief means P ∩ T = φ                                   The inverse of p ⇒ q is ~ p ⇒ ~ q
            i.e. there is no common element between P and T.
                                                                                    i.e., If a number is not a prime then it is not odd.
37.   (c) Some teenagers are not dreamers means teenagers
          which are not dreamers.
38.   (c) All mothers are women.
            M ⊆ W.
39.   (b) ~ (q ∨ ~ ( p ∧ r)) = ~ q ∧ (~ (~ ( p ∧ r)) = ~ q ∧ ( p ∧ r) .

40.   (c)
             p   ~p     p⇒~p          ~p⇒ p   ( p ⇒ ~ p ) ∧ (~ p ⇒ p )

             T    F        F           T                  F
             F    T        T           F                  F
            Clearly, ( p ⇒ ~ p ) ∧ (~ p ⇒ p ) is a contradiction.

41.   (c) p ⇒ q ≡ ~ p ∨ q ∴ ~ ( p ⇒ q ) ≡ p ∧ ~ q .

42.   (c) Contrapositive of p ⇒ q is ~ q ⇒ ~ p

            ∴ contrapositive of ( p ∨ q ) ⇒ r is

            ~ r ⇒ ~ ( p ∨ q ) i.e. ~ r ⇒ (~ p ∧ ~ q ) .

43.   (a) p ⇒ q is false only when p is true and q is false.

            ∴ p ⇒ q is false when p is true and q ∨ r is false,
            and q ∨ r is false when both q and r are false.

            Hence truth values of p, q and r are respectively T, F,
            F.
44.   (b) ( p ⇒ q ) ∧ (q ⇒ p ) means p ⇔ q .

45.   (b) p ⇒ q is logically equivalent to ~ q ⇒ ~ p

            ∴ ( p ⇒ q ) ⇔ (~ q ⇒ ~ p ) is a tautology but not a
            contradiction.




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Description: AIEEE IIT-JEE CBSE STUDY MATERIAL MATHEMATICS SAMPLE PAPERS TEST PAPER KEY SOLUTIONS ANSWERS QUESTIONS KEY CONCEPTS