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Thinking Mathematically Chapter 3 Logic Thinking Mathematically Section 1 Statements, Negations, and Quantified Statements Statements A statement is a sentence that is either true or false, but not both at the same time. Examples “Miami is a city in Florida” is a true statement. “Two plus two equals five” is a false statement. “Today is Friday” is a statement that is sometimes true and sometimes false, but never both true and false at the same time. “Go to the grocery store” is not a statement. It is a command and is neither true nor false. Negation of Statements The negation of a statement is another statement that has the opposite truth value. That is, when a statement is true its negation is false and when the statement is false its negation is true. Examples “Miami is not a city in Florida” is the negation of the statement “Miami is a city in Florida”. “Two plus two is not equal to five” is the negation of the statement “Two plus two equals five”. Be careful! “Two plus two equals four” is a true statement but it is not the negation of “Two plus two equals five”. Symbolic Statements 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. Negating Quantified Statements The negation of a statement about all objects is not all. Not all can often be expressed by some are not. The negation of a statement about some objects is not some. Not some can often be expressed by none or not any. Statement Negation of statement All A are B No A are B Some A are B Some A are not B Thinking Mathematically Section 2 Compound Statements and Connectives Compound Statements Simple statements can be connected with and, either … or, if … then, or if and only if. These more complicated statements are called compound statements. 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 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 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. The Two Meanings of Or • Compound statements involving the word or are a bit harder to understand. • There are two meanings: • 1. Exclusive or: At a Chinese Restaurant, the lunch menu says: "Choose one from column A or one from column B." • You can only choose one item, so you are not allowed to choose from both columns. The Two Meanings of Or • Compound statements involving the word or are a bit harder to understand. • There are two meanings: • 2. Inclusive or: "A good teacher should be either smart or funny." Here, you are implying that it's possible for a good teacher to be both smart AND funny, but at least one of these characteristics should be true. The Two Meanings of Or In this class, we will use the word or to be the inclusive or: one or the other or both! If ... then Statements 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 When two statements are represented by p and q the compound if and only if statement is: p q. p: The word I'm thinking of is "set". q: The word has 464 meanings. p q: The word I'm thinking of is "set" if and only if the word has 464 meanings. ~q ~p: The word I'm thinking of does not have 464 meanings if and only if the word is not "set". Translating sentences into symbolic expressions • Words such as and, but, or, either, neither, if, then, and so on are used in ordinary sentences and have meanings in logic. • Example: – He went to the store but didn't buy anything. Translating sentences into symbolic expressions He went to the store but didn't buy anything. He went to the store and didn't buy anything. He went to the store yet he didn't buy anything. Although he went to the store, he didn't buy anything. He went to the store ; nevertheless, he didn't buy anything. Translating symbolic expressions into sentences • If he goes home, it is not true that he missed his curfew or that he will be late for work tomorrow. – p: he goes home – q: he missed his curfew – r: he will be late for work tomorrow. p ~( q ∨ r ) Symbolic Logic Statements of Logic Name Symbolic Form Meaning Negation ~p not Conjunction pq and Disjunction pq or Conditional pq if..then Biconditional pq is the same as Thinking Mathematically Section 3 Truth Tables for Negation, Conjunction, and Disjunction Truth Tables - Negation If a statement is true then its negation is false and 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 The conjunction of two statements is true only when both of them are true. This can be represented in the form of a truth table in the following way. p q pq T T T T F F F T F F F F Truth Tables - Disjunction The disjunction of two statements is false only when both of them are false. This can be represented in the form of a truth table in the following way. p q pq T T T T F T F T T F F F Constructing a Truth Table 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 does this look familiar? Constructing a Truth Table Construct a truth table for ( p ~q) (~p q ) p q ~p ~q (p ~q) (~p q) (p ~q) (~p q) T T F F F F F T F F T T F T F T T F F T T F F T T F F F Section Quiz Construct a truth table for (p q) (~p ~q) p q ~p ~q (p q) (~p ~q) (p q) (~p ~q) T T F F T F T T F F T F F F F T T F F F F F F T T F T T Section Quiz Construct a truth table for (p q) ~r p q r ~r (p q) (p q) ~r T T T F T T T T F T T T T F T F F F T F F T F T F T T F F F F T F T F T F F T F F F F F F T F T Section Quiz • For each of the following 10 expressions, write down the value of the expression: • For example 2. F ∨F F ∧ 3. ~(~F)F F 10. ∨ ∨F ∨ 1. T TF∧∨T F T • 5. 4. 9. 6. 8. 7. • =T Solutions 1. T ∧ F = F 2. F ∨ F = F 3. T ∧ T ∧ T = T 4. F ∨ T =T 5. T ∧ F =F 6. ~(~F) = F 7. F ∧ F =F 8. T∨F∨F =T 9. T ∧ T =T 10. F ∨ F =F Section Quiz • Construct a truth table for ~p ∨ q p q ~q ~p ∨ q p q ~q ~p ∨ q p q ~q ~p ∨ q T T F T T T T T T T F T T F F T T F F T T F T T F T T F F T T T F T F F F F T T F F F F F F T T p q ~q ~p ∨ q T T F T T F F T F T T F F F T T Thinking Mathematically Section 4 Truth Tables for the Conditional and Biconditional Conditional Statements The truth table for implication (→) is p q p→ q T T T T F F F T T F F T This one is difficult to understand. Let's try anyway. Conditional Statements The truth table for implication (→) is p q p→ q T T T T F F F T T F F T Suppose I say, "If it rains tomorrow, I'll be carrying an umbrella." p: it is raining tomorrow q: I'm carrying an umbrella Conditional Statements The truth table for implication (→) is p q p→ q T T T No. I told T F F the truth F T T F F T The next day, it's raining. You look out your window and there I am, keeping dry under an umbrella. Did I lie? Conditional Statements The truth table for implication (→) is p q p→ q T T T I sure did! T F F F T T F F T The next day, it's raining. You look out your window and there I am, soaking wet without an umbrella. Did I lie? Conditional Statements The truth table for implication (→) is p q p→ q T T T Not really. T F F F T T I never said F F T what I'd do if it didn't rain. The next day, it's sunny. You look out your window and there I am, holding an umbrella. Did I lie? Conditional Statements The truth table for implication (→) is p q p→ q T T T Not really. T F F F T T I never said F F T what I'd do if it didn't rain. The next day, it's sunny. You look out your window and there I am, without an umbrella. Did I lie? Conditional Statements The truth table for implication (→) is p q p→ q T T T T F F F T T F F T p is called the precedent (or hypothesis) q is called the consequent (or consequence) An implication is false if and only if the precedent is true and the consequent is false. Biconditional Statements The truth table for the biconditional, "if and only if", (↔) is p q p↔q T T T T F F F T F F F T This operator can be viewed in two ways: - A two-way implication: p → q and q → p Biconditional Statements The truth table for the biconditional, "if and only if", (↔) is p q p↔q T T T T F F F T F F F T This operator can be viewed in two ways: - The concept of "has exactly the same truth table as" or "is equivalent to" Biconditional Statements • Look at the truth tables below: p q q↔p p q pq qp pq /\ qp T T T T T T T T T F F T F F T F F T F F T T F F F F T F F T T T Section Quiz • For each of the following 10 expressions, write down the value of the expression: • For example 1. T TT TTF 10. ∨ ∧F 2. F ⟷ ~F • ⟷ F 3. ~(~T)FF 5. 6. 9. 8. 4. 7. • =T Solutions 1. T F = F 2. F ⟷ F = T 3. T ∧ T = T 4. F T = T 5. T ∧ ~F = T 6. ~(~T) = T 7. F F =T 8. T⟷F = F 9. T ∧ T =T 10. T F = F Thinking Mathematically Section 5 Equivalent Statements, Variations of Conditional Statements, and DeMorgan’s Laws Equivalent Statements We saw at the end of a previous lecture that the truth table for (p /\ ~q) \/ q was the same as the one for p \/ q . Whenever two statements have the same truth table values, we say they are equivalent. Equivalent Statements Two statements are equivalent if they have the same truth values. p q pq p q ~p ~pq T T T T T F T T F T T F F T F T T F T T T F F F F F T F So the statements pq and ~pq are equivalent. They mean the same thing! Equivalent Statements Whoa!!!! The statements p q and ~p q mean the same thing?????????? Equivalent Statements The statements p q and ~p q are equivalent. They mean do the same thing! •Why? • In order for p q to be true, either p or q or maybe both have to true. That means that if p is false, we would need for q to be true. • If p is false, then q is true. • If ~p is true, then q is true. • If not p then q. • ~p q The Contrapositive The statements p q and ~q ~p are equivalent. They are contrapositives. p q pq p q ~p ~q ~q ~q ~p ~q ~p T T T T T F F T T F F T F F TT F F T T F T T F T F F T F F F F T T TT T For example, "If it rains tomorrow, I'll take my umbrella" means exactly the same thing as "If I don't take my umbrella tomorrow, it's because it isn't raining." The Contrapositive The statements p q and ~q ~p are equivalent. They are contrapositives. For example, "If it rains tomorrow, I'll take my umbrella" means exactly the same thing as "If I don't take my umbrella tomorrow, it's because it isn't raining." Think about it. The original statement says that on the condition that it's raining, I'll definitely take an umbrella. So if you see me not carrying an umbrella, then it can't be raining! Converse and Inverse q p is the converse of p q. ~p ~q is the inverse of p q. The converse and inverse are contrapositives of each other and are equivalent. p q qp p q ~p ~q ~p~q T T T T T F F T T F T T F F T T F T F F T T F F F F T F F T T T Conditional Statements Let p and q be statements. Name Symbolic Form Conditional pq Converse qp Inverse ~p ~q Contrapositive ~q ~p Conditional Statements Conditional: If it rains tomorrow, I'll take my umbrella. Converse: If I have my umbrella tomorrow, it will be raining. Inverse: If it doesn't rain tomorrow, I won't take my umbrella. Contrapositive: If I don't have my umbrella tomorrow, it won't be raining. DeMorgan's Laws • The statements ~(p ∧ q) and ~p ∨ ~q are equivalent. This is one of two DeMorgan's Laws. • ~(p ∧ q) says "It is false that p and q are true" • ~p ∨ ~q says "Either p is not true or q is not true" • These are the same because if it is false that p and q are both true, then either p has to be false or q has to be false. DeMorgan's Laws • The statements ~(p ∧ q) and ~p ∨ ~q are equivalent. This is one of two DeMorgan's Laws. p q pq ~(pq) p q ~p ~q ~p~q T T T F T T F F F T F F T T F F T T F T F T F T T F T F F F T F F T T T DeMorgan's Laws • The statement ~(p\/q) and ~p/\~q are equivalent. This is the other DeMorgan's Law. p q pq ~(pq) p q ~p ~q ~p~q T T T F T T F F F T F T F T F F T F F T T F F T T F F F F F T F F T T T Another equivalence • The statements p→q and ~p q are equivalent. p q p→q p q ~p ~p\/q T T T T T F T T F F T F F F F T T F T T T F F T T F F T Another equivalence • The statements p→q and ~p q are equivalent. • If p is true (p), that means that ~p is false. • So, in order for ~p q to be true, if p is true (in other words, ~p is false), the other half of ~p q, namely q, must be true. If p then q. Thinking Mathematically Section 6 Arguments and Truth Tables Arguments There are two parts to an argument. The first part consists of a collection of statements called the premises. The second part is a statement called the conclusion. Symbolically the argument is the conditional statement whose if part is the conjunction of all the premises and whose then part is the conclusion. An Example of an Argument p q pq If I get an A on the final I will pass the course. p I got an A on the final. q I will pass the course The argument is “If I get an A on the final I will pass the course and I got an A on the final. Therefore I will pass the course.” [(p q) /\ p] q An Example of an Argument pq If I get an A on the final I will pass the course. p I got an A on the final. q I will pass the course The argument says that getting an A on the final will assure me of passing the course. Since I did get an A, the professor has to give me a passing grade (or else he lied to me.) We can show that this argument makes sense by showing that the corresponding truth table is a tautology. A tautology is a logical statement that is always true; its truth table has T on all lines. Valid Arguments Valid arguments are tautologies. That is, they are always true. pq If I get an A on the final I will pass the course. p I got an A on the final. q I will pass the course [(p q) /\ p] q p q pq (pq)p q [(pq)p]q tautology !!! T T T T T T this is a valid T F F F F T argument !!! F T T F T T F F T F F T Another Example of an Argument pq If I get an A on the final I will pass the course. q I passed the course !!! p I must have gotten an A on the final [(p q) /\ q] p p q pq (pq)q p [(pq)q]p not a tautology !!! T T T T T T T F F F T T not a valid F T T T F F argument !!! F F T F F T Evaluating Real Arguments • Step 1: Assign variables (p, q, r, …) to each basic statement made in the argument. • Step 2: For each premise in the argument, write the corresponding logical expression. • Step 3: Write the corresponding logical expression for the conclusion. • Step 4: Connect all the premise expressions with . Connect that big expression with the conclusion expression with . • Step 5: Construct the truth table for the overall expression. • Step 6: If the expression is a tautology, it is a valid argument. Another Example: • Here's a real-world argument. Is it a valid argument? – If I go out, I'll go to the store. – If I go to the store, I will buy some chips. – I'm not going out. – Therefore, I won't be buying chips. Another Example: p q • If I go out, I'll go to the store. • If I go to the store, I will buy some chips. • I'm not going out. r • Therefore, I won't be buying chips. Step 1: Assign variables (p, q, r, …) to each basic statement made in the argument. Another Example: • If I go out, I'll go to the store. q p • If I go to the store, I will buy some chips. qr • I'm not going out. ~p • Therefore, I won't be buying chips. ~r [(p q) (q r) ~p] ~r For each premise in the argument, Step 2:3: Write the corresponding logical write the Step Connect all the premise expressions with . Step 4: corresponding logical expression. expression big expression with Connect that for the conclusion. the conclusion expression with . Step 6: If the expression is atable for the overall expression. Step 5: Construct the truth tautology, it is a valid argument. p q r p q q r ~p [(p q) ~r [(p q) (q r) ~p] ~r (q r) ~p T T T T T F F F T T T F T F F F T T T F T F T F F F T T F F F T F F T T F T T T T T T F F F T F T F T F T F F T T T T T F F F F T T T T T not a tautology. not valid Another Example: • If it is not sunny, it is either overcast or it's the evening. • If it's the evening, it can't be overcast. • If it's overcast, I will wear a raincoat. • It's not the evening. • Therefore, I am wearing a raincoat. Another Example: • p: it is sunny. • q: it is overcast. • r: it is the evening. • s: I am wearing a raincoat. Another Example: • If it is not sunny, it is either overcast or it's the evening. ~p(qr) • If it's the evening, it can't be overcast. r~q • If it's overcast, I will wear a raincoat. qs • It's not the evening. ~r • Therefore, I am wearing a raincoat. s • [(~p(qr)) (r~q) (qs) ~r] s First half of truth table. Other half is when p is false p q r s ~p(qr) r~q qs ~r [(~p(qr)) (r~q) [(~p(qr)) (qs) ~r] (r~q) (qs) ~r] s T T T T T F T F F T T T T F T F F F F T T T F T T T T T T T T T F F T T F T F T T F T T T T T F F T T F T F T T T F F T T F F T T T T T T T T F F F T T T T T F not a tautology. not valid ~p is F Section Quiz • I'll either get my hair cut or I won't go outside. • I'm not getting my hair cut • I won't go outside. • Is this a valid argument? p: get haircut Section Quiz q: go outside • I'll either get my hair cut or I won't go outside. p ~q • I'm not getting my hair cut. ~p • Therefore, I won't go outside. ~q [(p ~q) ~p] ~q • Is this a valid argument? Section Quiz Is this a valid argument? yes ~q p q ~p ~q p ~q (p ~q)~p [(p~q)~p]~q T T F F T F F T T F F T T F T T F T T F F F F T F F T T T T T T Section Quiz • Which of the following is a tautology? – a) p ~p – b) p ~p – c) p → ~p – d) p ↔ ~p

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