A Set is a collection of distinct and defined elements. Sets are represented by using French braces {} with commas to separate the elements in a Set.

Examples of Sets:
• The set of all points on a particular line.
• The set of all lines in a particular plane.
• A set can also contain elements which are themselves sets. For example, a set may contain $4, 5, \{ 6, 7 \}$.

There are three basic Set Operations.
  • Intersection of Sets
  • Union of Sets
  • Complement of Sets
Intersection of two sets:
The Intersection of two or more sets is a set containing only the common elements among all the sets under consideration. The intersection operation is denoted by the symbol $\cap$. The Intersection of the sets $A$ and $B$ is expressed as $A \cap B$.


Union of sets:
The Union of two or more sets is a set that contains all the elements of all the sets under consideration. The union of sets is denoted by the symbol $\cup$. The Union of sets $C$ and $D$ is expressed by $C \cup D$.

Complement of Sets:
The Complement of a set is a set that contains only the elements which are not in the given set but are contained in the universal set. The Complement of a set is denoted by $A$'.

 The Complement of an Universal Set is a Null Set. That is, $U$' = $\phi$

 The Complement of a Null Set is an Universal Set. That is, $\phi$' = $U$

There are various types of sets:
• Finite Set
• Infinite Set
• Null Set
• Singleton Set
• Equivalent Sets
• Equal Sets

Finite Set

$A$ set is said to be finite if it contains only finite number of elements.

Let $A$ = $\{ 1, 2, 3, 4, 5 \}$
    $B$ = $\{ a, b, c, d, e, f \}$

Here, $A$ contains $5$ elements and $B$ contains $6$ elements. So, $A$ and $B$ are finite sets 

Examples of Finite Set

• The set $\{ 2, 4, 6 \}$ Is a finite set, as it contains only $3$ elements.

• If $A$ is the set of all days in a week, then $A$ is a finite set containing $7$ elements.

Infinite Set

$A$ set is said to be infinite if it contains an infinite number of elements.

Let $C$ = {number of men living in different parts of the world} 

It is difficult to find the number of elements in $C$. But, it is a definite number, may be quite a big number. And so, $C$ is an infinite set.

Examples of Infinite Set

• The set of all even numbers is an infinite set.

• The set of points on a particular straight line is an infinite set.

• The sets $N$ (Natural Numbers), $Z$ (Integers), $Q$ (Rational Numbers), $R$ (Real Numbers) and $C$ (Complex Numbers) are all infinite sets, where:

1) $N$ = $\{ 1, 2, 3, 4, ....................... \}$

2) $Z$ = $\{ .............., -3, -2, -1, 0, 1, 2, 3, .......... \}$

3) $Q$ = $\{$ $\frac{(p)}{(q)}$ : $p, q$ in $Z, q \notin 0 \}$

4) $R$ = $\{ x : " x$ is either a rational number or an irrational number " $\}$

5) $C$ = $\{ x + iy : x, y$ in $R, i$ = $\sqrt{-1} \}$

Null Set

A set is said to be a null set if it does not contain any element. A null set is also called as an empty set or void set. A null set is denoted by $\phi$.

Therefore, $\phi$ = $\{ \}$

The set $\{ 0 \}$ is not a null set, because this set contains one element "$0$".

Examples of Null Set

• Let $A$ = $\{ x : x\ \epsilon\ N, 2 < x <3 \}$. $A$ does not contain any element , because there is no natural number between $2$ and $3$.

• Let $B$ = $\{ x : x\ \epsilon\ Q, 2 < x < 3 \}$. $B$ is not a null set, because rational numbers like $\frac{5}{2}$, $\frac{7}{3}$, $\frac{9}{4}$, $\frac{11}{5}$, ....... are all elements of the set $B$.

Singleton Set

A set is said to be a singleton set, if it contains only one element.

Examples of the Singleton Set

• The set $\{ 7 \}$ , $\{ -15 \}$ are singleton sets.

• $\{ x : x + 4 = 0, x\ \epsilon\ Z \}$ is a singleton set, because this set contains only one integer namely, $-4$.

Equivalent Sets

Two sets $A$ and $B$ are said to be equivalent sets if the elements of $A$ can be paired with the elements of $B$, so that each element of $A$ corresponds exactly to one element of $B$, and each element of $B$ there corresponds exactly to one element of $A$.

Examples of Equivalent Sets

• The sets $\{ a, b, c \}$ and $\{ 4, 7, 10 \}$ are equivalent.

• The sets $\{ w, x, y, z \}$ and $\{ 1, 2, 3, 4 \}$ are equivalent.

Equal Sets

Two sets are said to be equal sets if every element of one set is in the other set and vice-versa. So, two sets are equal, if $x$ in $A \Rightarrow x$ in $B$ and $x\ \epsilon\ B \Rightarrow x\ \epsilon\ A$. If sets $A$ and $B$ are not equal, then we write $A \neq B$

Examples of Equal Sets

Let $A$ = $\{ x : x\ \epsilon\ N,\ 2 \leq x \leq 6 \}$ and $B$ = $\{ 2, 3, 4, 5, 6 \}$ , then $A$ = $B$

Let $A$ = $\{ x : x\ \epsilon\ N, 10 < x < 11 \}$ and $B$ = $\{ 10.5 \}$ , then $A \neq B$ since $10.5\ \epsilon\ A$
Cardinality is a measure of a size of a set or simply the total number of elements in a set.

Examples of Cardinal Number of a Set


  • Cardinality of a set of vowels is 5 as it has 5 elements (a, e ,i , o, u).
  • Cardinality of a Set of the number of months in a year is 12 as it has 12 elements (January, February, ......., December).

Notation: Cardinality of a set is denoted by |A| or #A. The Cardinality of a set of natural numbers is denoted by N0 and that of the Real numbers is C.
Given below are some solved examples in the set theory.

Example 1:

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, B = (1, 3, 6, 7, 8} and C = {3, 7}, find A ∩ B, A ∪ C, B ∩ A′, B ∩ C′

Solution:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 4, 6, 8, 10}
B = (1, 3, 6, 7, 8}
C = {3, 7}

A ∩ B = {6, 8}
A ∪ C = {2, 3, 4, 6, 7, 8, 10}
B ∩ A′ = {1, 3, 7}
B ∩ C′ = {1, 6, 8}

Example 2:

If U = {Pencil, Pen, Eraser, Notebook}, P = {Pencil, Notebook} and Q = {Pen, Eraser}, find P ∪ Q, P ∩ Q, P ∪ Q', P ∩ Q'

Solution:

U = {Pencil, Pen, Eraser, Notebook}
P = {Pencil, Notebook}
Q = {Pen, Eraser}

P ∪ Q = {Pencil, Notebook, Pen, Eraser} = U
P ∩ Q = ∅
P ∪ Q' = {Pen, Eraser, Pencil, Notebook}
P ∩ Q' = ∅

Example 3:

At a breakfast buffet, 93 people preferred coffee as a beverage, 47 people preferred juice, 25 preferred both coffee and juice. If each person prefers atleast one of the beverages, then how many people visited the buffet?

Solution:

Let A be the set of people who prefer coffee and B be the set of people who prefer juice.

n(A) = 93, n(B) = 47, n(A ∩ B) = 25

n(AUB) = n(A) + n(B) - n(A ∩ B)

Plugging-in all the values,

n(AUB) = 93 + 47 - 25
n(AUB) = 115

Hence, the number of people who visited the buffet is 115.

Given below are some laws of set theory:

1 ¬(¬A) = A Law of Double Complement
2 ¬(A ∪ B) = ¬A ∩ ¬B
¬(A ∩ B) = ¬A ∪ ¬B
DeMorgan's Laws
3 A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutative Laws
4 A ∪ (B ∪ C) = (A ∪ B) ∪ C
A ∩ (B ∩ C) = (A ∩ B) ∩ C
Associative Laws
5 A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Distributive Laws
6 A ∪ A = A
A ∩ A = A
Idempotent Laws
7 A ∪ ∅ = A
A ∩ U = A
Identity Laws
8 A ∪ ¬A = U
A ∩ ¬A = ∅
Inverse Laws
9 A ∪ U = U
A ∩ ∅ = ∅
Domination Laws
10 A ∪ (A ∩ B) = A
A ∩ (A ∪ B) = A
Absorption Laws
11 A - B = A ∩ ¬B Definition of Set Difference
12 A Δ B = (A ∪ B) - (A ∩ B) Definition of Symmetric Difference

A Set is called as a subset of another set if all it's elements are contained in another set. The symbol used for the subset is ⊂. The set A is a subset of the set B and is expressed as A ⊂ B. Each non empty set has at least two subsets - Empty Set (or Null Set) and the set itself. The number of subsets of a set is calculated by the formula 2n, where n is the number of elements of the Set.

Empty Set or Null Set is represented by either {} or Φ (read as phi).


A set A is said to be a proper subset of a set B, if A is a subset of B and A is not equal to B. If A is a proper subset of B, then we write A ⊂ B. If A is a proper subset of B, then B must have at least one element which is not in A.

Example:

Let B be the set of closed objects as shown below:
proper set
Let us consider the set A as follows:
proper set example
Then, clearly every object in A is an object in B. Hence, A ⊂ B.
Two sets A and B are called disjoint sets if there is no element which is both in A and B. Disjoint sets can be also defined as those sets which do not overlap or are not duplicated. Let us consider two sets as follows:

Let A be the set of triangles as shown
disjoint set

Let B be the set of arrows as shown
disjoint set example
Here, the set A and B have no elements in common. So, A ∩ B = Φ. Hence, such sets are called as disjoint sets.

Example

The set of intervals { [1, 3], [2,5], (7,9)} is not disjoint, since [1, 3] is overlapping [2,5]. Disjoint sets are also said to be mutually exclusive or independent.

Disjoint sets can be represented by using venn-diagram as:
disjoint set using venn diagram

Mutually Disjoint Sets


Set A1, A2, A3, ......, An are mutually disjoint if Ai ∩ Aj = Φ for i ≠ j. A partition of set A is a collection of subsets Ai of A such that Ai ≠ Φ , A1, A2, A3, .........., An are mutually disjoint and ∪ Ai = A

For example, let the universal set be U = {1, 2, 3, 4, 5}
and let its subsets be A = {1, 2, 3} and B = {4, 5}.
Then, A ∩ B = Φ and A ∪ B = U. So, A and B are partitions of U