2 Conditional and Biconditional Propositions
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1 18 FUNDAMENTALS OF MATHEMATICAL LOGIC 2 Conditional and Biconditional Propositions Let p and q be propositions. The implication p! q is the proposition that is false only when p is true and q is false; otherwise it is true. p is called the hypothesis and q is called the conclusion. The connective! is called the conditional connective. Example 2.1 Construct the truth table of the implication p! q: The truth table is Example 2.2 Show that p! qp _ q. p q p! q T T T T F F F T T F F T p q p p! q p _ q T T F T T T F F F F F T T T T F F T T T It follows from the previous example that the proposition p! q is always true if the hypothesis p is false, regardless of the truth value of q: We say that p! q is true by default or vacuously true. In terms of words the proposition p! q also reads: (a) if p then q: (b) p implies q: (c) p is a su cient condition for q: (d) q is a necessary condition for p: (e) p only if q:
2 2 CONDITIONAL AND BICONDITIONAL PROPOSITIONS 19 Example 2.3 Use the if-then form to rewrite the statement \I am on time for work if I catch the 8:05 bus." If I catch the 8:05 bus then I am on time for work In propositional functions that involve the connectives ; ^; _; and! the order of operations is that is performed rst and! is performed last. Example 2.4 a. Show that (p! q) p^ q: b. Find the negation of the statement \ If my car is in the repair shop, then I cannot go to class." a. We use De Morgan's laws as follows. (p! q) ( p _ q) ( p)^ q p^ q: b. \My car is in the repair shop and I can get to class." The converse of p! q is the proposition q! p: The opposite or in-verse of p! q is the proposition p! q: The contrapositive of p! q is the proposition q! p: Example 2.5 Find the converse, opposite, and the contrapositive of the implication: \ If today is Thursday, then I have a test today." The converse: If I have a test today then today is Thursday. The opposite: If today is not Thursday then I don't have a test today. The contrapositive: If I don't have a test today then today is not Thursday Example 2.6 Show that p! q q! p:
3 20 FUNDAMENTALS OF MATHEMATICAL LOGIC We use De Morgan's laws as follows. p! q Example 2.7 Using truth tables show the following: a. p! q 6 q! p b. p! q 6 p! q a. It su ces to show that p _ q 6 q _ p: p _ q (p^ q) ( q ^ p) q_ p q_ p q! p p q p q p _ q q _ p T T F F T T T F F T F 6= T F T T F T 6= F F F T T T T b. We will show that p _ q 6 p_ q: Example 2.8 p q p q p _ q p_ q T T F F T T T F F T F 6= T F T T F T 6= F F F T T T T Show that q! p p! q We use De Morgan's laws as follows. q! p q_ p ( q ^ p) (p^ q) p_ q p _ q p! q
4 2 CONDITIONAL AND BICONDITIONAL PROPOSITIONS 21 The biconditional proposition of p and q, denoted by p $ q, is the propositional function that is true when both p and q have the same truth values and false if p and q have opposite truth values. Also reads, \p if and only if q" or \p is a necessary and su cient condition for q." Example 2.9 Construct the truth table for p $ q: p q p $ q T T T T F F F T F F F T Example 2.10 Show that the biconditional proposition of p and q is logically equivalent to the conjunction of the conditional propositions p! q and q! p: p q p! q q! p p $ q (p! q) ^ (q! p) T T T T T T T F F T F F F T T F F F F F T T T T The order of operations for the ve logical connectives is as follows: ^; _ in any order. 3.!; $ in any order.
5 22 FUNDAMENTALS OF MATHEMATICAL LOGIC Review Problems Problem 2.1 Rewrite the following proposition in if then form: \ This loop will repeat exactly N times if it does not contain a stop or a go to." Problem 2.2 Construct the truth table for the proposition: p _ q! r: Problem 2.3 Construct the truth table for the proposition: (p! r) $ (q! r): Problem 2.4 Write negations for each of the following propositions. (Assume that all variables represent xed quantities or entities, as appropriate.) a. If P is a square, then P is a rectangle. b. If today is Thanksgiving, then tomorrow is Friday. c. If r is rational, then the decimal expansion of r is repeating. d. If n is prime, then n is odd or n is 2. e. If x 0; then x > 0 or x = 0: f. If Tom is Ann's father, then Jim is her uncle and Sue is her aunt. g. If n is divisible by 6, then n is divisible by 2 and n is divisible by 3. Problem 2.5 Write the contrapositives for the propositions of Problem 2.4. Problem 2.6 Write the converse and inverse for the propositions of Problem 2.4. Problem 2.7 Use the contrapositive to rewrite the proposition \ The Cubs will win the penant only if they win tomorrow's game" in if then form in two ways. Problem 2.8 Rewrite the proposition : \Catching the 8:05 bus is su cient condition for my being on time for work" in if then form. Problem 2.9 Use the contrapositive to rewrite the proposition \being divisible by 3 is a necessary condition for this number to be divisible by 9" in if then form in two ways.
6 2 CONDITIONAL AND BICONDITIONAL PROPOSITIONS 23 Problem 2.10 Rewrite the proposition \A su cient condition for Hal's team to win the championship is that it wins the rest of the games" in if then form. Problem 2.11 Rewrite the proposition \A necessary condition for this computer program to be correct is that it not produce error messages during translation" in if then form.
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