# Statements and negations (3.1) p. 88-93 Compound statements and

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```					Statements and negations (3.1) p. 88-93
Compound statements and connectives (3.2) p. 94-101

OBJECTIVES:

Express   statements using symbols

Form   the negation of statements

Form   compound statements using connectives
Statements – a sentence that is either true or false,
but not both simultaneously, p. 89
Examples
5<9
17 < 9
11 > 5
All two digit numbers are greater than 1 digit numbers

Negation- a statement with an opposite “truth.
When a statement is true its negation is false and
when the statement is false its negation is true. p. 89
Statements - “Symbolism”
Just as x can be used as a name for a number, a
symbol such as p can be used as a name for a
statement.
When p is used as a name for a statement the symbols
~p are used as a name for the negation of p.
Examples
Let p stand for “Miami is a city in Florida.”
Then ~p is the statement “Miami is not a city in
Florida.”
“Quantified” Statements
A “quantified” statement is one that says something
about “all”, “some”, or “none” of the objects in a
collection.
Examples
“All students in the college are taking history.”
“Some students are taking mathematics.”
“No students are taking both mathematics and
history.”
“Equivalent” Statements
In any language there are many ways to say the same
thing. The different linguistic constructions of a
statement are considered equivalent.
Example
“All students in the college are taking history.”
“Every student in the college is taking history.”
Example
“Some students are taking mathematics.”
“At least one student is taking mathematics.”
Negating Quantified Statements
The negation of a statement about “all” objects is
“not all”. “Not all” can often be expressed by “some
are not.”
Examples
p : All students in the college are taking history.
~p : Some students in the college are not taking
history.
Negating Quantified Statements
The negation of a statement about “some” objects is
“not some”. “Not some” can often be expressed by
“none” or “not any.”
Examples
p : Some students are taking mathematics.
~p : None of the students are taking mathematics.
“Compound” Statements, 94
Simple statements can be connected with “and”, “Either
… or”, “If … then”, or “if and only if.” These more
complicated statements are called “compound.”
Examples
“Miami is a city in Florida” is a true statement.
“Atlanta is a city in Florida” is a false statement.
“Either Miami is a city in Florida or Atlanta is a city in
Florida” is a compound statement that is true.
“Miami is a city in Florida and Atlanta is a city in
Florida” is a compound statement that is false.
“And” Statements, p. 95
When two statements are represented by p and
q the compound “and” statement is p /\ q.
p: Harvard is a college.
q: Disney World is a college.
p/\q: Harvard is a college and Disney World is
a college.
p/\~q: Harvard is a college and Disney World is
not a college.
“Either ... or” Statements, p. 96

When two statements are represented by p and q the
compound “Either ... or” statement is p\/q.
p: The bill receives majority approval.
q: The bill becomes a law.
p\/q: The bill receives majority approval or the bill
becomes a law.
p\/ ~q: The bill receives majority approval or the
bill does not become a law.
“If ... then” Statements, p. 96
When two statements are represented by p and q the
compound “If ... then” statement is: p  q.
p: Ed is a poet.
q: Ed is a writer.
p  q: If Ed is a poet, then Ed is a writer.
q  p: If Ed is a writer, then Ed is a poet.
~q  ~p: If Ed is not a writer, then Ed is not a Poet
“If and only if” Statements, p.98

When two statements are represented by p and q the
compound “if and only if” statement is: p  q.
p: The word is set.
q: The word has 464 meanings.
p  q: The word is set if and only if the word has
464 meanings.
~q  ~p: The word does not have 464 meanings if
and only if the word is not set.
Symbolic Logic, p. 99
Statements of Logic
Name            Symbolic Form
Negation             ~p
Conjunction          p/\q
Disjunction          p\/q
Conditional          pq
Biconditional        p q
“Truth Tables” – Negation, p. 103
If a statement is true then its negation is false.
If the statement is false then its negation is true.
This can be represented in the form of a table
called a “truth table.”
p     ~p
T     F
F     T
“Truth Tables” – Conjunction, p.104
The conjunction of two statements is true
only when both of them are true.

p      q     pq
T      T      T
T      F      F
F      T      F
F      F      F
“Truth Tables” – Disjunction, p.106
The disjunction of two statements is false
only when both of them are false.

p     q     pq
T     T      T
T     F      T
F     T      T
F     F      F
Constructing a Truth Table, p. 107

Construct a truth table for ((p/\~q)\/q).

p    q      ~q     (p~ q) (p~ q)q
T    T      F         F        T
T    F      T         T        T
F    T      F         F        T
F    F      T         F        F
HOMEWORK

Officehours                M-F 9:00-10:15
Beach Bldg Room 113 or by appointment
Work p.93 #1-30 odd
Work p.101