SETS

A set is a well-defined collection of objects.
The objects in a set are called elements or members.
Example:
A = {2, 4, 6, 8}
Therefore:
2 ∈ A
4 ∈ A
5 ∉ A
The symbol ∈ means "is an element of", while ∉ means "is not an element of".

Set Notation
Sets are usually represented using capital letters.
Example:
B = {a, b, c, d}
The elements are written inside curly brackets { }.
A set can also be described using a rule.
Example:
A = {x : x is an even number less than 10}
This means:
A is the set of x such that x is an even number less than 10.
Therefore:
A = {2, 4, 6, 8}

Types of Sets
1. Null/Empty Set
A set with no element.
Symbol:
∅ or { }
Example:
A = {months with 32 days}
A is a null set.
2. Singleton Set
A set containing only one element.
Example:
B = {7}
3. Finite Set
A set with a limited number of elements.
Example:
C = {1, 2, 3, 4, 5}
4. Infinite Set
A set whose elements continue without end.
Example:
D = {1, 2, 3, 4, 5, ...}

Subsets
A set A is a subset of B if every element of A is also found in B.
Symbol:
A ⊆ B
Example:
A = {1, 2}
B = {1, 2, 3, 4}
Therefore:
A ⊆ B

Universal Set
The universal set, represented by U, contains all the objects being considered in a particular situation.
Example:
If we are considering numbers from 1 to 10:
U = {1,2,3,4,5,6,7,8,9,10}

Power Set
The power set of A is the set containing all possible subsets of A.
It is represented by P(A).
Example:
If:
A = {1, 2}
Then:
P(A) = {∅, {1}, {2}, {1,2}}

Important Rule
If a set has n elements, its power set has:
2ⁿ subsets
Therefore, if A has 3 elements:
Number of subsets = 2³ = 8

Set Operations
1. Union
The union of A and B contains everything in A or B or both.
Symbol:
A ∪ B
Example:
A = {1,2,3}
B = {3,4,5}
Therefore:
A ∪ B = {1,2,3,4,5}
2. Intersection
The intersection contains elements that are common to both sets.
Symbol:
A ∩ B
Therefore:
A ∩ B = {3}
3. Complement
The complement of A contains elements in the universal set that are not in A.
Symbol:
A′ or Aᶜ
Example:
U = {1,2,3,4,5}
A = {1,2,3}
Therefore:
A′ = {4,5}

Number of Elements in a Set
The number of elements in a set is called its cardinality.
It is written as n(A) or |A|.
Example:
A = {2,4,6,8}
Therefore:
n(A) = 4
Two-Set Formula
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
This prevents the common elements from being counted twice.

Venn Diagrams
A Venn diagram uses circles inside a rectangle to represent sets.
Rectangle = Universal set
Circle = Set
Overlapping area = Intersection
Combined circles = Union
Area outside a set but inside U = Complement
Example
In a class:
20 students like Mathematics.
15 like Science.
8 like both.
Then:
n(M ∪ S) = 20 + 15 − 8
= 27 students