Study Guide for Exam 1
Study Guide for Exam 1 COMP 11500
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This 4 page Study Guide was uploaded by Daniel Akimchuk on Friday February 19, 2016. The Study Guide belongs to COMP 11500 at Ithaca College taught by Ali Erkan in Spring 2016. Since its upload, it has received 88 views. For similar materials see Discrete Structures for Computer Science in ComputerScienence at Ithaca College.
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Date Created: 02/19/16
Exam 1 Study Guide – Sections 1.1-1.9 and 2.1-2.4 • Definitions o Proposition: The most basic logical element. Has a truth value of either true or false. o Compound Proposition: A proposition made from conjoining propositions with logical operators. o Tautology: A compound proposition that is true regardless of the truth values of the individual propositions. (i∨e¬p) o Contradiction: A compound proposition that is false regardless of the truth values of the individual propositions. (i∧e¬p). p o Predicate: A logical statement whose truth value is a function of one or more variables. Written “P(x),” where x is the variable it relies on. o Free Variable: A variable which is not within a quantif. r o Bound Variable: A variable within a quantifier. o Theorem: A statement that can be proven. o Proof: A series of step that logically follow each other, whose final step is the theorem being proven. o Axiom: Used in proofs. A statement assumed to be true, or a previously proven theorem. o Direct Proof: A proof where you assume the hypothesis and the con clusion is proven as a direct result. o Proof by Contrapositive: A proof where you prove a statement true by proving the contrapositive true. If you want to prove pàq, prove ¬qà¬p. o Proof by Contradiction: A proof where you assume the opposite of the theorem you are trying to prove, and try to find a logical inconsistency. If you are trying to prove t, assume ¬t. If you are trying to prove pàq, assum∧ ¬q. Also known as an indirect proof. • Logical Operators and Quantifiers o Conjunction: “p and q,” written p∧ q. o p q p∧ q 1 1 1 1 0 0 0 1 0 0 0 0 o Disjunction: “p or q,” written ∨ q. Also known as the inclusive or. o p q p∨ q 1 1 1 1 0 1 0 1 1 0 0 0 o Exclusive Or: “p or q,” written p⊕q. o p q p⊕q 1 1 0 1 0 1 0 1 1 0 0 0 o Negation: “not p,” written ¬p. o p ¬p 1 0 0 1 o Conditional: “if p, then q,” “p implies q,” “q if p,” “p only if q,” “q is necessary for p,” “p is sufficient for q,” written pàq. o p q pàq 1 1 1 1 0 0 0 1 1 0 0 1 o Biconditional: “p if and only if q,” “p iff q,” written p↔q. True whenever p and q have the same truth value. o p q p↔q 1 1 1 1 0 0 0 1 0 0 0 1 o Universal Quantifier: “for all x, P(x)” “P is true for all x in its do∀ax ,” written P(x). o Existential Quantifier: “there exists an x in which P(x) is tru∃x P(x).ten • Laws of Propositional Logic o Idempotent Laws o p ∨ p ≡ p o p ∧ p ≡ p o Associative Laws o (p ∨ q)∨ r ≡ ∨( q∨ r) o (p ∧ q)∧ r ≡ ∧( q∧ r) o Commutative Laws o p ∨ q ≡ ∨ p o p ∧ q ≡ ∧ p o Distributive Laws o p ∨ (∧ r) ≡ (∨ q∧ (p∨ r) 2 o p ∧(q∨r) ≡ ∧q)∨ (∧ r) o Identity Laws o p ∨False ≡ p o p ∨True ≡ True o p ∧False ≡ False o p ∧True ≡ p o Double Negation Law o ¬¬p ≡ p o Complement Laws o p ∨¬p ≡ True o p ∧¬p ≡ False o ¬True ≡ False o ¬False ≡ True o DeMorgan’s Laws o ¬(p∨ q) ≡ ∧¬q o ¬(p∧ q) ≡ ∨¬q o Absorption Laws o p ∨(p∧q) ≡ p o p ∧(p∨q) ≡ p o Conditional Identities o pàq ≡ ¬p∨ q o p↔q ≡ (pàq)∧ (qàp) o DeMorgan’s Law for Quantified Statements o ¬ ∀x P(x)∃x ¬P(x) o ¬ ∃x P(x)∀x ¬P(x) • Other Notes o The order of precedence for logical operators and quantifiers is as follows: 1. Quantifiers (∃x and ∀x) 2. Negation (¬) 3. Conjunctio∧) 4. Disjuncti∨n) 5. Exclusive Or (⊕) 6. Conditional (à) 7. Biconditional (↔) o Just like in algebra, anything in parentheses goes first. n o When making truth tables, compound propositions with n variables will have 2 rows of possible truth value combinations . o To make sure you get all possible combinations, count in binary, starting at 0 . o Given the statement “pàq:” 1. Converse: qàp 2. Inverse: ¬pà¬q 3 3. Contrapositive: ¬qà¬p o A statement and its contrapositive are logically equivalent (pàq ≡ ¬qà¬p). o A statement’s converse and inverse are logically equivalent (qàp ≡ ¬pà¬q). o A statement without a free variable is a proposition. o Example of a direct proof: Theorem: Every positive integer is less than or equal to its square . Proof: 1 Let x be an integer greater than 0. (x > 0) 2 Since x is an integer greater than 0, x ≥ 1. 3 Multiply both sides by x: x • x ≥ 1 • x 2 4 So, x ≥ x. o When proving a universal statement, start with a general statement that assumes no more than the theorem does. 4
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