Sets: Types, Representation, Symbols, Properties, Examples (2024)

Operations on Sets

The various operations on sets are given below:

Venn Diagrams:Most of the relationships between sets can be interpreted utilizing diagrams recognized as Venn diagrams. Venn diagrams are identified after the English logician, John Venn. These diagrams consist of rectangles and closed curves and regular circles. The universal set is designated by a rectangle and its subsets are denoted by circles. We will get a clear idea about the rectangle and circle concept in the operations below.

Union of Sets: Let A and B be two sets. The union of sets A and B is the set of all those components which belong to either A or B or both A and B. The union of A and B is indicated by A ∪ B.

i.e A ∪ B = {x : x ∈ A or x ∈ B}

⇒Example: If A = {2, 3, 5} and B = {2, 3, 5, 7}. Then A ∪ B = {2, 3, 5, 7}.

The Venn diagram for the union of any two sets is shown below for a better understanding:

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Intersection of Sets: Let A and B be two sets. The intersection of sets A and B is the set of all those components which are present in both sets A and B.

⇒The intersection of A and B is indicated by A ∩ B i.e A ∩ B = {x: x ∈ A and x ∈ B}

⇒Example: If A = {2, 3, 5} and B = {2, 3, 5, 7}. Then A ∩ B = {2, 3, 5}

The Venn diagram for intersection is shown below to have a clear idea:

Disjoint Sets:Any two sets say A and B are said to be disjoint if A ∩ B = ϕ i.e they don’t have any common component.

Example: If A = {1, 2, 3} and B = {4, 5, 6, 7}. Then A ∩ B = ϕ

The Venn diagram illustration of disjoint sets is shown below:

Difference Between two Sets

Let A and B be two sets. The difference between A and B expressed as (A – B) or A \ B, is the set of all those elements of A which are not present in B.

i.e A – B = {x : x ∈ A and x∉ B}

Example: If A = {1, 2, 3, 5, 7} and B = {2, 3, 5}. Then A – B = {1} and B – A = {7}

The Venn diagram representation of the difference between the two sets is shown below:

Symmetric Difference between two Sets

Let A and B be two sets. The symmetric difference of sets A and B is the set (A – B) ∪ (B – A) and is denoted as A Δ B.

i.e A Δ B = (A – B) ∪ (B – A)

⇒Example: If A = {1, 2, 3, 5, 7} and B = {2, 3, 5}. Then A – B = {1}, B – A = {7} and A Δ B = {1, 7}.

The Venn diagram illustration of the symmetric difference between two sets is presented below:

Complement of a Set

Let A be any set. Then complement of set A is denoted by A’ or Ac such that A’ = {x: x ∉ A} = U – A, where U is the universal set.

The Venn diagram illustration of the complement of a set is shown below:

The next concept in line is the laws regarding set theory, the laws and properties always act as a time saver element in solving the numerical during the examination. So let us look at the list of laws of Sets Math one by one.

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Laws of Algebra of Sets

Read about the Laws of Algebra of Sets in the table below:

Sets Formulas

Consider A, B and C to be three finite sets and U representing the finite universal set, then;

1. n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
2. n (A ∪ B) = n (A) + n (B) ⇔ A ∩ B = ϕ
3. n (A – B) = n (A) – n (A ∩ B) = n (A ∩ B’)
4. n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n(A ∩ B) – n (B ∩ C) – n (A ∩ C) + n (A ∩ B ∩ C)
5. n (A’ ∪ B’) = n [(A ∩ B)’] = n (U) – n (A ∩ B)
6. n (A’ ∩ B’) = n [(A ∪ B)’] = n (U) – n (A ∪ B)
7. n (A Δ B) = n (A) + n (B) – 2 n (A ∩ B)
8. n (A’) = n (U) – n (A)

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