Chapter 7 - Sets and Venn Diagrams
Z
Integers sign Rational numbers (can be written as a fraction) Z = { ..., -3, -2, -1, 0, 1, 2, 3, ...}
⊂
Proper subset
Q
Rational numbers written in the form p/q where p and q are integers and q ≠ 0 Infinite, can't list
R
Real numbers sign Infinite, can't list
U
the universal set the only possible elements under consideration for that particular problem
Subsets in Venn Diagrams
B ⊆ A B inside A
Intersections in Venn Diagrams
B∩A ∩ = center spot in between A and B
Unions in Venn Diagrams
B∪A Includes A, B, and the intersection between A and B
Venn Diagram
Consists of a universal set U represented by a rectangle. Sets within U are usually represented by circles.
∈
Element sign EX: 2 ∈ A → "two is an element of set A"
∩
Intersection sign
N
Natural numbers sign N = {0, 1 , 2, 3, 4, 5, 6, 7, ...}
Z-
Negative integers sign Z- = {..., -4, -3, -2, -1}
∉
Not an element sign EX: y ∉ V → "y is not an element of set V"
P⊂Q
P is a proper subset of Q P is a subset of Q but Q contains at least one element that is not in P. EX: P = {17, 21, 35} Q = {9, 17, 18, 21, 24, 34, 35} So... P⊂Q AND P⊆Q
P⊆Q
P is a subset of Q Every element of P is also an element of Q. EX: Q = {1, 2, 3} Subsets of Q = {1} {1, 2} {2} {3} {2, 3} {1, 3} {1, 2, 3} {∅}
Z+
Positive integers sign Z+ = {1, 2, 3, 4, ...}
Set Builder Notation
EX: A = {x | -3 ≤ x ≤ 5, x ∈ Z} "The set of all x such that x is an integer between -3 and 5 inclusive." n(A) = 9 EX: B = {x | x > 0, x ∈ R} "The set of all x such that x is a real number greater than zero." n(B) = ∞
'
the complement of a set
A'
the complement of set A the set of all elements of U which are not in A EX: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 3, 6, 8} A' = {2, 4, 5, 7, 9, 10}
Properties
A∩U = A A∪U = U A∩A' = ∅ A∪A' = U n(A) + n(A') = n(U)
P∩Q
"The intersection of P and Q" The set of all elements that are in both P and Q EX: P = {0, 1, 3, 4, 6, 8, 9, 13} Q = {-2, 1, 2, 5, 6, 7, 10, 12} P∩Q = {1, 6}
Set
A collection of numbers or objects. EX: A = {0, 1, 2, 3, 4, 5} V = {a, e, i, o, u} P = {2, 4, 6, 8, 10, ...}
Finite Set
A set that has a limited amount of numbers that ends. EX: A = {0, 1, 2, 3, 4, 5} V = {a, e, i, o, u}
Infinite Set
A set that has an unlimited amount of numbers that never end. EX: P = {2, 4, 6, 8, 10, ...}
n(P)
Represents the number of elements in set P EX: n(∅) = 0 n({1, 3, 5, 7}) = 4
Disjoint/Mutually Exclusive
Sets that have no elements in common
⊆
Subset sign
∅
The empty set Is a subset of ALL sets
Elements
The objects in a set. (AKA Members)
P∪Q
The union of P and Q The set of all elements that are in either P or in Q or both EX: P = {1, 2, 3, 4} Q = {2, 3, 4, 5} P∪Q = {1, 2, 3, 4, 5}
∪
Union