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# Chapter 2 Section 1-2 MATH174

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This 3 page Class Notes was uploaded by Kaleigh Kelley on Wednesday February 10, 2016. The Class Notes belongs to MATH174 at University of South Carolina Upstate taught by Chris Woodard in Winter 2016. Since its upload, it has received 39 views. For similar materials see Discrete Mathematics in Mathematics (M) at University of South Carolina Upstate.

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Date Created: 02/10/16

DISCRETE CHAPTER 2 SECTION 2.1-2.2 Set- collection of “objects” called elements or members We use capital letters or sets o For example, S= {5,7,9} o 5 € S o {55,999,7777} = {5,7,9} Empty set is also known as a null set or void set o Represented as { } or Ø Cardinality of a set- the number of distinct elements of a set o The cardinal number of a set o Represented as |A| The natural number- {1,2,3,4,5,6,7,8} o Represented as o Roster Notation: {1,2,3,4,5,6,7,8} o Set Buidler Notation: {X € /N | x ≤ 100} The set of Intergers {-2, -1, 0, 1, 2,} o Represented as o {X € : 1 ≤ X ≤ 100} The Rational Numbers- numbers that can be written as the ratio of 2 integers a,b where b ≠ 0. o Represented as o {x : x = a/b , a, b, € , and b ≠ 0} Subset- a set of A is called a subset of set B o Represented as The empty set- {} is a subset of every set o If every element of A also belongs to B o Example: A = juniors and B= students, then Proper- set A is a proper subset of set B o Represented as A ⊂ B o If but A ≠ B Number of subsets of a set with n elements = 2^n All but one subset is proper (the set itself is not a proper subset) o Number of proper subsets = 2^n1 Venn Diagram- displays relationship between subsets Powerset- a set S is the set of all subsets of S o Represented by P(S) o P(S) = {A | A ⊆ S} o If |A| = n |P | (A) | = 2^n o Example: B= {4,5} P (B) = { {4,5} , {4}, {5}, Ø} SECTION 2 SET OPERATORS AND THEIR PROPERTIES Union- of two sets A and B is U o In one or both sets o Example: A U B= {X : X € A or X € B} Intersection- of two sets A and B is A∩B o A∩B = { X: X € A and X € B} Example of Union and Intersections: A= {1,2,3,4,5} B= {0,2,4,6} A∩B = {2,4} A U B= {0,1,2,3,4,5,6} Laws Communitive- A∩B = B∩A Associative= A U (B U C)= (A U B)UC Distributive= A∩(B U C) = (A∩B) U (A∩C) Disjoint- no elements in common o A and B are disjoint if and only if A∩B= Ø Pairwise disjoint- if an 2 sets are disjoint Difference- (A – B) o Elements in A but not B o {X: X€B and X ∉ B} Symmetric Difference o Things in A but not in B o A ⊕ B = (A – B) U (B- A) Set Compliment- of is Ā o Ā = {X: X € U and X ∉ A} DeMorgan’s Law o A U B = A∩B and A∩B= A U B

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