1 Logical equivalence

When proving a proposition in mathematics it is often useful to look at a logical variation of the proposition in question that “means the same thing”. What does “meaning the same thing” mean? For our purposes, in keeping with our “meaning is truth, truth meaning” mantra, it will mean having the same truth-conditions. This is the notion of logical equivalence. Definition 1.1. Two (possibly compound) logical propositions are logically equivalent if they have the same truth tables.

Comment 1.1. More specifically, to show two propositions P1 and P2 are logically equivalent, make a truth table with P1 and P2 above the last two columns. The two are logically equivalent when these last two columns are identical.

Comment 1.2. The fact that those columns are identical means that P1 and P2 have the same truth value in every possible circumstance.

1.1 Contrapositive, converse and inverse Definition 1.2. Given the implication P ⇒ Q, the implication (Not Q)⇒ (Not P ) is called its contrapos- itive. Let’s show that the implication P ⇒ Q and its contrapositive ¬Q ⇒ ¬P are logically equivalent. P Q P ⇒ Q ¬Q ⇒ ¬P T T T T T F F F Since the two propositions P ⇒ Q and ¬Q ⇒ ¬P have the same truth F T T T F F T T values for each possible truth value of P and Q, we see that they are logically equivalent. Recall that the converse of P ⇒ Q is the implication Q ⇒ P . Definition 1.3. The inverse of P ⇒ Q is the contrapositive of its converse: namely, the implication ¬P ⇒ ¬Q. Since any implication is logically equivalent to its contrapositive, we know that the converse Q ⇒ P and the inverse ¬P ⇒ ¬Q are logically equivalent. In all we have four different implications. P ⇒ Q ¬Q ⇒ ¬P Q ⇒ P ¬P ⇒ ¬Q. Implications lying in the same row are logically equivalent. Implications in different rows are not logically equivalent.

1.2 Examples Example. Show that Not (P or Q) is logically equivalent to Not(P ) and Not(Q). P Q Not(P or Q) Not(P ) and Not(Q) T T F F T F F F F T F F F F T T

Example. Show that P ⇒ Q is logically equivalent to (P ⇒ Q) and (R or Not(R)).

1 P Q R P ⇒ Q (P ⇒ Q) and (R or Not(R)) T T T T T T T F T T T F T F F T F F T T F T T T T F T F T T F F T F F F F F T T

1.3 Logical tautology and logical contradiction Definition 1.4. A proposition is a logical tautology if it is always true (no matter what the truth values of its component propositions). Similarly, a proposition is a logical contradiction (or an absurdity) if it is always false (no matter what the truth values of its component propositions). P P or not(P ) Example (Logical tautology). P or not(P ). T T F T

P P and not(P ) Example (Logical contradiction). P and not P . T F F F

P Q (P or not P )⇒(Q and not Q) T T F Example. (P or not P )⇒(Q and not Q). T F F F T F F F F

2 Proof method

Many of the propositions you will be asked to prove (or disprove) will take the form of an implication

P ⇒ Q

or an equivalence P ⇔ Q. Example. Prove: if n2 is an odd integer, then n is an odd integer. Example. Prove: n2 is an odd integer if and only if n is an odd integer.

Our truth tables for implication and equivalence indicate how we should prove such statements.

2.1 Implication According to our truth tables, to prove directly that P ⇒ Q is true, we need only show that if P is true, then Q is true; this is because when P is false, the implication is vacuously true. Thus to prove P ⇒ Q is true, we assume that P is true, and use this to show that Q is true. Recall that P ⇒ Q is logically equivalent to the contrapositive ¬Q ⇒ ¬P . This suggests an indirect way of proving P ⇒ Q: namely, we can prove its contrapositive. Logical equivalence guarantees that this is a valid proof method: the implication is true exactly when the contrapositive is true; so if we can show the contrapositive is true, we know the original implication is true too!

2 Example. Let n be an integer. We will prove indirectly that if n2 is an odd, then n is odd. • The contrapositive of this is ‘If n is not odd, then n2 is not odd’. Since ‘not odd’ is the same as ‘even’, we have the statement ‘If n is even, then n2 is even’. • Now prove the contrapositive. Assume n is even. Then we can write n = 2r for some r. But then n2 = 4r2 = 2(2r2) = 2s is even. This proves the contrapositive, and hence the original implication.

2.2 Equivalence When we first defined what P ⇔ Q means, we said that this equivalence is true if P ⇒ Q is true and the converse Q ⇒ P is true. This is in fact a consequence of the truth table for equivalence. So one way of proving P ⇔ Q is to prove the two implications P ⇒ Q and Q ⇒ P . Example. Let n be an integer. Prove that n2 is odd if and only if n is odd. • We must prove TWO implications, P ⇒ Q and Q ⇒ P . • We have already proved P ⇒ Q. • To prove Q ⇒ P , assume n is odd. Then n2 = n · n is also odd since an odd times an odd is odd. This proves Q ⇒ P . • Since both implications are true, the if and only if statement is true. Since the converse Q ⇒ P is logically equivalent to the inverse ¬P ⇒ ¬Q, another way of proving the equivalence P ⇔ Q is to prove the implication P ⇒ Q and its inverse ¬P ⇒ ¬Q. In summation we have two different ways of proving P ⇔ Q: 1. Prove P ⇒ Q and Q ⇒ P , or 2. Prove P ⇒ Q and ¬P ⇒ ¬Q.

2.3 Proof by contradiction We end with a description of proof by contradiction. This method sets out to prove a proposition P by assuming it is false and deriving a contradiction. Example. Prove by contradiction that there is no greatest even integer. Proof. Suppose by contradiction that there is a greatest even integer. Call this integer n. Since n is even, so is n + 2 (even plus even is even). But n + 2 is greater than n and n + 2 is even, contradicting the fact that n is the greatest even integer. Thus our original assumption must be false; i.e., there can be no greatest integer. What are we really doing when we prove a proposition P by contradiction, and why is this valid? In essence we prove an implication of the form (¬P ) ⇒ (Some false statement). Call the false statement in question Q. Since (¬P ) ⇒ Q is true and Q is false, our truth table for implication tells us that ¬P must be false. But this means that P is true, which is what we wanted to show. Comment 2.1. The notion of proof by contradiction is often confused with the indirect method for proving an implication we discussed earlier. These are distinct proof methods. 1. We prove the implication P ⇒ Q indirectly by proving the contrapositive ¬Q ⇒ ¬P . 2. We prove the proposition P by contradiction by proving an implication of the form (¬P ) ⇒ (Some false statement).

3