Section 9 Chapter 51 - Sets
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Set
An unordered collection of values where each value appears at most once
Methods to define a set (3)
• Listing each number • Set comprehension • Compact representation
Symbol for empty set
∅
Symbol for rational number set
Q
Definition for the set of real numbers
The set of all possible real world quantities
Finite set
A set where each element can be counted off by natural numbers up to a particular number
Cardinality of a finite set
The number of elements in the set
Infinite set
A set where you cannot count off each element by natural numbers up to a particular number. It is a set which is not a finite set
Countably infinite set
A set that can be counted off by the natural numbers
Countable set
A set which can be counted off against a subset of the natural numbers
Set comprehension
A method of defining a set where you specify specify some function of n and for which values that function should be run for
Symbol for “such that”
|
Symbol for membership
∈
Symbol for and
∧
Write the set {1,4,9,16,25} using set comprehension
A = {n^2 | n < 6 ∧ n ∈ N}
Compact representation
A method of defining a set using string-like properties
Write the set {01, 0011, 000111…} using compact representation
A = {0^n 1^n | n > 0 ∧ n ∈ N}
Empty set
The set with no elements
Cartesian product of two sets
The set of all ordered pairs (a,b) where a is a member of A and b is a member of B
{1,2,3}x{3,2}
{(1,3), (1,2), (2,3), (2, 2), (3,3), (3,2)}
Subset
A is a subset of B if every element in A is also in B
Proper subset
A is a proper subset of B if every element in A is also in B and A does not equal B
Symbol to show A is a subset of B
A⊆B
Symbol to show A is a proper subset of B
A⊂B