1. Conditional Statements Lecture 2 Section 1.2 Fri, Jan 20, 2006

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.” The idea is that the truth of p implies the truth of 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. pq p  q TTT TFF FTT FFT

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). It is false only if it is raining and I am not 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. p  qp  q q  pq  p p  qp  q q  pq  p converses inverses contrapositives

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). pq p  q TTT TFF FTF FFT

9. Exclusive-Or The statement p  q is the exclusive-or of p and q. p  q is defined by pq p  q TTF TFT FTT FFF

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 is also logically equivalent to (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 pqp | q TTF TFT FTT FFT

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 pq p  q TTF TFF FTF FFT

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).