1. 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. 2.2 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

3. 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

4. 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

5. 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

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

7. 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

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. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

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. 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