Discrete Structures

1. Discrete Structures Chapter 2: The Logic of Compound Statements 2.2 Conditional Statements … hypothetical reasoning implies the subordination of the real to the realm of the possible… – Jean Piaget, 1896 – 1980 2.2 Conditional Statements

2. Logic • The dean has announced that If the mathematics department gets an additional $40,000, then it will hire one new faculty member. The above proposition is called a conditional proposition. Why? 2.2 Conditional Statements

3. Conditional • Definition • If p and q are the statement variables, the conditional of q by p is “If p then q” or “p implies q” and is denoted pq. It is false when p is true and q is false. Otherwise, it is true. • We call p the hypothesis (or antecedent) of the conditional. • q is the conclusion (or consequent) of the conditional. 2.2 Conditional Statements

4. Example – pg. 49 #2 • Rewrite the statement in if-then form. • I am on time for work if I catch the 8:05 am bus. 2.2 Conditional Statements

5. Conditional Truth Table • The truth value for the conditional is summarized in the truth table on the right. 2.1 Logical Forms and Equivalences

6. Order of Operations • According to the order of operations, • First  • Second  • Third  • Fourth  • Fifth  2.2 Conditional Statements

7. Example – pg. 49 #5 • Construct a truth table for the statement form p q  q conclusion hypothesis 2.2 Conditional Statements

8. Negation of a Conditional Statement • By definition, pq is false iff its hypothesis, p, is true and its conclusion, q, is false. It follows that (p  q)  p  q Proof: 2.2 Conditional Statements

9. Example – pg. 49 # 20 b • Write the negations for each of the following statements. • If today is New Year’s Eve, then tomorrow is January. 2.2 Conditional Statements

10. Contrapositive • Definition • The contrapositive of a conditional statement of the form “If p then q” is If q then p Symbolically, the contrapositive of p  q is q  p 2.2 Conditional Statements

11. Example – pg. 49 # 22 b • Write the contrapositive for the following statement. • If today is New Year’s Eve, then tomorrow is January. 2.2 Conditional Statements

12. Converse & Inverse • Definition • Suppose a conditional statement of the form “If p then q” is given, • The converse is “If q then p.” • The inverse is “If p then not q.” 2.2 Conditional Statements

13. Example – pg. 49 # 23 b • Write the converse and inverse for each statement: • If today is New Year’s Eve, then tomorrow is January. 2.2 Conditional Statements

14. NOTE! • A conditional statement and its converse are not logically equivalent. • A conditional statement and its inverse are not logically equivalent. • The converse and the inverse of a conditional statement are logically equivalent to each other. 2.2 Conditional Statements

15. Only If • If p and q are statements, ponly if q means “if not q then not p or “if p then q.” 2.2 Conditional Statements

16. Biconditional - iff • Given the statement variables p and q, the biconditional of p and q is “piffq” denoted pq. It is true if both p and q have the same truth values and is false if p and q have opposite truth values. 2.2 Conditional Statements

17. Biconditional Truth Table • The truth value for the biconditional is summarized in the truth table on the right. 2.1 Logical Forms and Equivalences

18. Example – pg. 50 # 32 • Rewrite the statements as a conjunction of two if-then statements. • This quadratic equation has two distinct real roots if, and only if, its discriminate is greater than zero. 2.2 Conditional Statements

19. Necessary and Sufficient Conditions • Definition • If r and s are statements: • r is a sufficient condition for s means “if r then s.” • r is a necessary condition for s means “if not r then not s.” 2.2 Conditional Statements

20. Example – pg. 50 # 41 • Rewrite the statement in if-then form. • Having two 45 angles is a sufficient condition for this triangle to be a right triangle. 2.2 Conditional Statements