Conditional Statements

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Presentation transcript:

Conditional Statements Lecture 3 Section 1.2 Mon, Jan 22, 2007

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.

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

Example: Conditional Statements “If it is Wednesday, then Discrete Math meets today.” This statement is true if Discrete Math meets today (whether or not it is Wednesday), and if it is not Wednesday (whether or not Discrete Math meets today). It is false only if it is Wednesday and Discrete Math does not meet today.

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

The Converse and the Inverse The converse of p  q is q  p. The inverse of p  q is p  q. p  q q  p p  q q  p

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

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

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

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

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

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)  (p  q)

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).

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

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).

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).

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

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).