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