COSI 29A Chapter 1.1: Logic

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Computer Science
Cherniack Mitch

Chapter 1.1 Outline Logic Pages 1 – 20 Propositions A declarative sentence that is either true or false. The negation of a proposition (p) is denoted by ¬p (aka “not p”). Ex: The negation of “Today is Friday” is “Today is not Friday”. p ∧ q is called the conjunction of p and q. p ∨ q (“p or q”) is called the disjunction of p and q. The exclusive disjunction of p and q is denoted p ⨁ q (either p or q is true but not both). Implications The implication p → q (if p, then q) is only false when p is true and q is false (if the condition p is met then q must be also met). Ex: “If you get 100%, then you will get an A.” This statement is only false if you get 100% (p) but you do not get an A (q). The proposition q → p is the converse of p → q. The proposition ¬p → ¬q is the inverse of p → q. The proposition ¬q → ¬p is the contrapositive of p → q. The contrapositive has the same truth value as p → q. Since they have the same truth value they are called equivalent. The converse and inverse are equivalent. The biconditional p ↔ q is true when both p and q have the same truth values. p ↔ q is equivalent to (p → q) ∧ (q → p). Precedence of Logical Operators Negation has precedence over conjunction, conjunction has precedence over
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