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Conditional Statements

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

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Conditional Statements

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  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.

  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qandqp are logically equivalent.

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

  7. Is this logical?

  8. The Biconditional • The statement p qis the biconditionalof p and q. • p q is logically equivalent to (p  q)  (q  p).

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

  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 bothpandq. • 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

  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 neitherpnorq. • 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

  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).

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