Chapter 2: The Logic of Compound Statements 2.2 Conditional Statements 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

2.2 Conditional Statements 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

2.2 Conditional Statements 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

2.2 Conditional Statements 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

Conditional Truth Table The truth value for the conditional is summarized in the truth table on the right. p q p  q T F 2.1 Logical Forms and Equivalences

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

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

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

2.2 Conditional Statements 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

2.2 Conditional Statements 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

2.2 Conditional Statements 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

2.2 Conditional Statements 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

2.2 Conditional Statements 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

2.2 Conditional Statements 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

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

2.2 Conditional Statements Biconditional - iff Given the statement variables p and q, the biconditional of p and q is “p iff q” 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

Biconditional Truth Table The truth value for the biconditional is summarized in the truth table on the right. p q p  q T F 2.1 Logical Forms and Equivalences

2.2 Conditional Statements 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

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

2.2 Conditional Statements 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