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# Discrete Chapter 2 Study Guide MATH174

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This 4 page Study Guide was uploaded by Kaleigh Kelley on Tuesday February 16, 2016. The Study Guide belongs to MATH174 at University of South Carolina Upstate taught by Chris Woodard in Winter 2016. Since its upload, it has received 79 views. For similar materials see Discrete Mathematics in Mathematics (M) at University of South Carolina Upstate.

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Date Created: 02/16/16

CHAPTER 2 STUDY GUIDE NOTES 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 CHAPTER 2.3- CARTESIAN PRODUCTS OF SETS The Cartesian product of A and B is A x B o A x B= {(a,b): a € A and b € B} o (a,b) is an ordered pair Example: A = {1,2} B= {x,y} A X B= {(1,x), (1,y), (2,x), (2,y) The cardinality is 4 If |A| =m and |B| = n then |AxB|= m*n o Options of A multiplied by options of B give us the cardinality of the Cartesian product CHAPTER 2.4- PARTITION Partition- a partition of a non-empty set A is a collection of nonempty subsets of A such that every element of A belongs to exactly one of the subsets o A={1,2,3,4,5} P={S S 1,S2} 3 {{1,2} {3,4} {5}} S1={1,2} S 2{3,4} S 3{5} Every set is a subset of itself PRACTICE PROBLEMS 1. For S={-3,-2,-1….2} list the elements of the sets A= {n€S: 0 < |n| <2} 2. 3. Suppose A = {1, 2, 3, 4, 5}. Determine if each statement is true or false. (a) {1}P(A) (e) {}P(A) (b) {{3}}P(A) (f) {2,4}AA (c) A (g) |P(A)| = 5 d) {}P(A) (h) (1,1)AA 3. Let A, B be sets. Prove the De Morgan laws: 1. A ∪ B = A ∩ B 4. Let A, B be sets. Prove P(A ∩ B) = P(A) ∩ P(B). 5. Let A be the set of students in a discrete math class at your university and B the set of students majoring in computer science in your university. Describe the studnets in each of the following sets. A) A U B E) A ⊕ B B) A ∩ B D) B - A C) A – B 6) For A = {1,2} and B={-1,0,1} and the Universal Set U= {-2,-1,0,1,2}, determine the following (a) S= {x,y} (b) S= {1} 7) For the set S= {1,2,….6} give an example of a partition P of S such that | P| = 3 8) Determine whether the following statements are true or false A) {x} € {{X}} B) {(X,Y)} C {X.Y} C) {x,{X}} C {{X},{{X}}}

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