1. Conditional Statements
Lecture 2
Section 1.2
Thu, Jan 13, 2005

2. The Conditional A conditional statement is a statement of the form
p  q
p is the hypothesis.
q is the conclusion.
Read p  q as “p implies q.”

3. Truth Table for the Conditional
p  q is true if p is false or q is true.
p  q is false if p is true and q is false. p q
p  q T F

4. Example: Conditional Statements
“If it is raining, then I am carrying an umbrella.”
This statement is true
when I am carrying an umbrella (whether or not it is raining), and
when it is not raining (whether or not I am carrying an umbrella).

5. The Contrapositive The contrapositive of p  q is q  p.
The statements p  q and q  p are logically equivalent.

6. The Converse and the Inverse
The converse of p  q is q  p.
The inverse of p  q is p  q.
converses
p  q
q  p
p  q
q  p
inverses
contra
positives

7. Is this logical?

8. The Biconditional The statement p  q is the biconditional of p and q.
p  q is logically equivalent to
(p  q)  (q  p). p q
p  q T F

9. Exclusive-Or The statement p  q is the exclusive-or of p and q.
p  q is defined by p q
p  q T F

10. Exclusive-Or p  q means “one or the other, but not both.”
p  q is logically equivalent to
(p  q)  (q  p)
p  q is also logically equivalent to
(p  q)
(p  q)  (q  p)

11. The NAND Operator The statement p | q means not both p and q.
The operator | is also called the Scheffer stroke or NAND.
NAND stands for “Not AND.”
p | q is logically equivalent to (p  q).

12. The NAND Operator
p | q is defined by p q
p | q T F

13. The NAND Operator
The three basic operators may be defined in terms of NAND.
p  p | p.
p  q  (p | q) | (p | q).
p  q  (p | p) | (q | q).

14. The NOR Operator The statement p  q means neither p nor q.
The operator  is also called the Pierce arrow or NOR.
NOR stands for “Not OR.”
p  q is logically equivalent to (p  q).

15. The NOR Operator
p  q is defined by p q
p  q T F

16. The NOR Operator
The three basic operators may be defined in terms of NOR.
p  p  p.
p  q  (p  q)  (p  q).
p  q  (p  p)  (q  q).