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Unformatted text preview: Discrete Mathematics 1 Discrete Mathematics Book: Discrete Mathematics by K. P. Bogart Topics Sets Logic Relations & functions Induction Recurrence relations Graphs Grades: First: 25% Second 25% Final 50% Discrete Mathematics 2 Discrete Mathematics Chapter 1 Sets & Statements Discrete Mathematics 3 Sets Set = a collection of distinct unordered objects Members of a set are called elements a ∈ A “a is an element of A” “a is a member of A” a ∉ A “a is not an element of A” How to determine a set Listing: Example: A = {1,3,5,7} The result of tossing three coines S = {HHH, HHT, HTH, THH, HTT,THT, TTH, TTT} Description Example: B = {x  x = 2k + 1, 0 < k < 3} Discrete Mathematics 4 One other way of describing sets: Truth sets The set of all values of x that make a symbolic statement p(x) true is called the truth set of the proposition p. The symbolic statements p(x) & q(x) are equivalent if they have the same truth sets. Example: in the set S above let p(x) stands for “having exactly x heads”, q(x) stands for having exactly x tail” Is p(x) is equivalent to q(x)? Discrete Mathematics 5 Finite and infinite sets Finite sets Examples: A = {1, 2, 3, 4} B = {x  x is an integer, 1 < x < 4} D = {dog, cat, horse} D = {dog, cat, horse} Infinite sets Examples: Z = {integers} = {…, 3, 2, 1, 0, 1, 2, 3,…} Natural numbers N = {0, 1, 2, 3, …} S={x x is a real number and 1 < x < 4} = [0, 4] Discrete Mathematics 6 Venn diagrams A Venn diagram provides a graphic view of sets and their operations: union, intersection, difference, symmetric difference and complements can be identified Discrete Mathematics 7 Some important sets The empty set ∅ has no elements. Also called null set or void set. Universal set: the set of all elements about which we make assertions. Examples: U = {all natural numbers} U = {all real numbers} U = {x x is a natural number and 1< x< 10} Discrete Mathematics 8 Cardinality Cardinality of a set A (in symbols A) is the number of elements in A Examples: If A = {1, 2, 3} then A = 3...
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.
 Winter '08
 Staff
 Logic, Sets

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