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

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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.
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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
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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.
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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
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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
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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.
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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.”
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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.
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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)
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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
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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.
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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.
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LOGICAL STRUCTURE Exercises: A. Let p,q,r be the following sentences p: John is at the office.
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