In our daily lives, we constantly group similar items together.
This process of collecting objects of the same type is the foundation of set theory.
Examples of natural sets around us include:
A football team (11 players)
A class of students
A pack of playing cards
A group of colors
In mathematics, set theory helps us study and work with such collections formally.
A set is a well-defined collection of distinct objects.
The individual objects in a set are called elements or members.
Well-defined means we can easily decide whether an item belongs to the set or not.
Set of even numbers less than 10 = {2, 4, 6, 8}
Set of colors = {red, blue, green}
A set with no elements is called:
Empty set
Null set
Denoted by ∅ or {}
Example:
Set of secondary school students who are 155 years old = ∅ (impossible).
Sets are represented using capital letters, while elements are written in braces {}.
The elements are listed one by one inside braces.
Examples:
A = {a, b, c}
B = {blue, yellow, green, black}
C = {2, 4, 6, 8, 10}
The number of elements in a set is called its cardinality and is written as n(A).
Examples:
n(A) = 3
n(B) = 4
n(C) = 5
∈ means “is an element of”
∉ means “is not an element of”
Example:
Since 2 is in C, write 2 ∈ C
9 ∉ C
A set whose elements can be counted or listed completely.
Examples:
{1, 3, 5}, {blue, pink}, {boy1, boy2, boy3}
A set whose elements cannot be completely listed.
Examples:
Natural numbers N = {1, 2, 3, 4, …}
Integers Z = {…, −3, −2, −1, 0, 1, 2, 3, …}
(Three dots … show the set continues endlessly.)
Another way to describe sets without listing all elements.
Example:
E={x:x∈N,x>6}E = \{x : x \in N, x > 6\}E={x:x∈N,x>6}
Read as:
Set E is the set of all natural numbers greater than 6
Roster form:
E = {7, 8, 9, 10, 11, …}
Two sets are equal if they contain exactly the same elements.
If A and B contain the same members, write A = B.
If every element of A is also an element of B, then A is a subset of B.
Written as A ⊆ B
If A is a subset of B but the sets are not equal, then A is a proper subset of B.
Write A ⊂ B
Meaning:
All elements of A are in B, but B has extra elements not in A.
The power set of A is the set of all possible subsets of A.
Symbol: P(A)
If A has k elements,
|P(A)| = 2ᵏ
Let A = {a, b, c}
P(A) = {
∅,
{a}, {b}, {c},
{a, b}, {a, c}, {b, c},
{a, b, c}
}
k = 3 → 2³ = 8 subsets
A universal set (U) contains all possible objects related to a particular discussion.
It changes depending on context.
Examples:
In discussing even and odd numbers from 1–10:
U = {1,2,3,4,5,6,7,8,9,10}
For colors of the rainbow:
U = {red, orange, yellow, green, blue, indigo, violet}
The complement of A, written A′, is the set of elements in U that are not in A.
U = {1,2,3,4,5,6,7,8,9}
A = {1,3,5,7,9}
A′ = {2,4,6,8}
Important relations:
A ∪ A′ = U
A ∩ A′ = ∅
The union, A ∪ B, is a set containing all elements in A or B or both.
A = {1, 2, 3, 4, 5}
B = {3, 4, 5, 6, 7}
A ∪ B = {1, 2, 3, 4, 5, 6, 7}
The intersection, A ∩ B, is a set containing elements common to both A and B.
A = {1, 3, 5, 8}
B = {3, 5, 7}
C = {2, 4, 6, 8}
(i) A ∩ B = {3, 5}
(ii) A ∪ B = {1, 3, 5, 7, 8}
(iii) B ∩ (A ∪ C)
A ∪ C = {1,2,3,4,5,6,8}
So, B ∩ (A ∪ C) = {3,5,7} ∩ {1,2,3,4,5,6,8} = {3, 5}