discrete-structures

# Q the math dept hires one new faculty member p q if

• Notes
• 100

This preview shows pages 14–27. Sign up to view the full content.

q: The Math Dept. hires one new faculty member p→ q: If the Math Dept. gets an additional Php 200,000, then it will hire one new faculty member.

This preview has intentionally blurred sections. Sign up to view the full version.

LOGICAL STRUCTURE p→ q if p then q p implies q (q is implied by p) whenever p, q (q whenever p) q unless ¬p p only if q (if not q then not p) ¬q implies ¬p p is a sufficient condition for q
LOGICAL STRUCTURE examples 1. If 3+3=7, then you are the pope. 2. If the Lakers win the NBA, then they sign Artest. A necessary condition for the Lakers to win the NBA is that they sign Artest. 3. If John takes calculus, then he passed algebra. John may take calculus only if he passed algebra.

This preview has intentionally blurred sections. Sign up to view the full version.

LOGICAL STRUCTURE Definition: The truth value of the conditional proposition p→ q is defined by the following truth table. p q p→ q T T T T F F F T T F F T
LOGICAL STRUCTURE Examples: 1. p: 1>2 False q: 4<8 True p→ q ? q→ p ? 2. Given p is true, q is false, r is true, find the truth value of: a. (p q)→r b. (p q)→¬r

This preview has intentionally blurred sections. Sign up to view the full version.

LOGICAL STRUCTURE NOTE: Converse: The proposition q→ p is the converse of the proposition p→ q. Inverse: The proposition ¬p→¬q is the inverse of the proposition p→ q. Contrapositive : The contrapositive (or transposition) of the conditional proposition p→ q is the proposition ¬q→¬p.
LOGICAL STRUCTURE Example: “The home team wins whenever it is raining” (“If it is raining, then the home team wins.”) Converse: “ If the home team wins, then it is raining.” Inverse: “ If it is not raining, then the home team does not win.” Contrapositive: “If the home team does not win, then it is not raining.”

This preview has intentionally blurred sections. Sign up to view the full version.

LOGICAL STRUCTURE Definition If p and q are propositions, the compound proposition p if and only if q is called a biconditional proposition and is denoted by p↔ q.
LOGICAL STRUCTURE p↔ q p is equivalent to q p iff q p is a sufficient and necessary condition for q p →q and q→ p (p implies q and q implies p)

This preview has intentionally blurred sections. Sign up to view the full version.

LOGICAL STRUCTURE Definition The truth value of the proposition p↔ q is defined by the following truth table. Example: 1<5 iff 2<8 p q p↔q T T T T F F F T F F F T
LOGICAL STRUCTURE Derfinition Suppose that the compound propositions P and Q are made up of the propositions p 1 ,p 2 ,…,p n . We say that P and Q are logically equivalent and write P ≡ Q , provided that given any truth values of p 1 ,p 2 ,…,p n , either P and Q are both true or P and Q are both false.

This preview has intentionally blurred sections. Sign up to view the full version.

LOGICAL STRUCTURE example: 1. De Morgan’s Laws for Logic ¬ (p q) ≡ ¬p ¬q 2 . ¬(p→q) ≡ p ¬q 3. State whether P ≡ Q a. P = p (¬q r) ; Q = p (q ¬r) b. P = (p→q)→r ; Q = p→(q→r) Note: The conditional proposition p→q and its contrapositive ¬q→¬p are logically equivalent.
LOGICAL STRUCTURE Exercises: A. Let p,q,r be the following sentences p: John is at the office.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.
• Winter '99
• AverrÃ³is
• Logic, logical structure

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern