University:
Montgomery College
Course:
MATH 120: Survey of College Mathematics
Academic year:

2024
Views:

186

Pages:

5
Author:

Nico Bryan

Section 7.2 Sets Definitions: A set is any collection of objects. Each object in a set is called a member or element of the set. A set without any elements is called the empty or null set. A finite set is a set that has a finite number of elements. An infinite set is a set with an infinite number of elements. Rule Method vs Listing Method for Sets Example: 1) Complete the table. Rule Method *{x| x is a weekend day} Listing Method {Saturday, Sunday} Finite or Infinite? {x| x2 = 4} {x| x is an odd counting number} *Read: “The set of x such that x is a weekend day.” Symbols & Notation ∈ is an element of 𝐴 ⊂ 𝐵 : Set A is a subset of set B means every element of A is an element of B. 𝐴 = 𝐵 : Set A is equal to set B if every element of A is an element of B AND every element of B is an element of A, that is, 𝐴 ⊂ 𝐵 AND 𝐵 ⊂ 𝐴. ∅ = { } is the empty set. Note: The empty set is a subset of every set. Examples: Let A = {-3, -1, 1, 3}, B = {3, -3, 1, -1}, C = {-3, -2, -1, 0, 1, 2, 3}. Determine if the following are true or false. 2) 3 ∈ A 7) A ⊂ C 3) 0 ∈ A 8) 4) A ⊂ B 9) A ≠ C 5) B ⊂ A 10) ∅ ⊂ A 6) A = B 11) ∅ ⊂ C C⊂A 12) ∅ ∈ A Example: 13) List all subsets of the set {a, b, c}. Universal Set, Intersection & Union U: The universal set is the set of all elements under consideration. 𝐴 ∩ 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}: A intersect B is a set of all elements in A AND B. 𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}: A union B is a set of all elements in A OR B (or both). 𝐴′ = {𝑥 ∉ 𝐴}: The complement of A is the set of elements in the universal set that are not in A. Examples: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {3, 6, 9} and B = {3, 4, 5, 6, 7}. Determine the following sets. Write using the listing method. 14) A ∩ B 15) A ∪ B 16) A’ 17) B’ Venn Diagrams 𝐴∩𝐵 𝐴∪𝐵 𝐴′ Examples: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {3, 6, 9} and B = {3, 4, 5, 6, 7}. Draw a Venn diagram and then answer the following: 18) A ∩ B 19) A ∪ B 20) A’ 21) B’ 22) (A ∩ B)’ 23) (A ∪ B)’ Examples: Use the Venn Diagram to determine the indicated number of elements. 24) n(U) 25) n(A ∩ B) 26) n(A) 27) n(B) 28) n(A ∪ B) 29) n(A’) 30) n(B’) 31) A survey was given to 100 randomly chosen students which included the following three questions and responses. Do you own a TV? Do you own a car? Do you own a TV and a car? 75 said yes 45 said yes 35 said yes Draw a Venn diagram and answer the following questions: a) How many students do not own a TV? b) How many students do not own a car? c) How many students do not own a car or a TV? d) How many students own a TV but not a car? e) How many students own a car but not a TV? f) How many students own either a TV or a car?