Conditional Statements

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Presentation transcript:

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

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.”

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 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).

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. converses p  q q  p p  q q  p inverses contra positives

Is this logical?

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)  (q  p)

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).