Introduction to logic a proposition is a statement

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Introduction to Logic A proposition is a statement that is either true or false. For example, “2 + 2 = 4” and “Donald Knuth is a faculty at Rutgers-Camden” are propositions, whereas “What time is it?”, x 2 < x + 40 are not propositions. We can construct compound propositions from simpler propositions by using some of the following connectives. Let p and q be arbitrary propositions. Negation: ˜ p (read as “not p ”) is the proposition that is true when p is false and vice-versa. Conjunction: p q (read as “ p and q ”) is the proposition that is true when both p and q are true. Disjunction: p q (read as “ p or q ”) is the proposition that is true when at least one of p or q is true. Exclusive Or: p q (read as “ p exclusive-or q ”) is the proposition that is true when exactly one of p and q is true is false otherwise. Implication: p q (read as “ p implies q ”) is the proposition that is false when p is true and q is false and is true otherwise.
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6 Lecture Outline January 10, 2011 The implication q p is called the converse of the implication p q . The implication ¬ p → ¬ q is called the inverse of p q . The implication ¬ q → ¬ p is the contrapositive of p q . p only if q means “if not q then not p”, or equivalently if p then q . Biconditional: p q (read as “ p if, and only if, q ”) is the proposition that is true if p and q have the same truth values and is false otherwise. “If and only if” is often abbreviated as iff. The following truth table makes the above definitions precise. p q ¬ p p q p q p q p q q p p q T T F T T F T T T T F F F T T F T F F T T F T T T F F F F T F F F T T T Necessary and Sufficient Conditions: For propositions p and q , p is a sufficient condition for q means that p q . p is a necessary condition for q means that ¬ p → ¬ q , or equivalently q p . Thus p is a necessary and sufficient condition for q means “ p iff q ”. Logical Equivalence Two compound propositions are logically equivalent if they always have the same truth value. Two statement p and q can be proved to be logically equivalent either with the aid of truth tables or using a sequence of previously derived logically equivalent statements. Example. Show that p q ≡ ¬ p q ≡ ¬ q → ¬ p . Solution. The truth table below proves the above equivalence. p q ¬ p ¬ q p q ¬ p q ¬ q → ¬ p T T F F T T T T F F T F F F F T T F T T T F F T T T T T Example. Show that p ≡ ¬ p C . p ¬ p C ¬ p C T F F T F T F F The above equivalence forms the basis of proofs by contradiction.
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January 10, 2011 Lecture Outline 7 The logic of Quantified Statements Consider the statement x < 15. We can denote such a statement by P ( x ), where P denotes the predicate “is less than 15” and x is the variable. This statement P ( x ) becomes a proposition when x is assigned a value. In the above example, P (8) is true while P (18) is false.
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