Venn Diagram is a Pictorial Representation of Relation Between Sets. In general, Venn Diagrams are represented using the Symbols Circles or Ovals. The Elements of the Sets are placed within the Circle. To help you understand the Relationship in Sets using Venn Diagram we have taken a few examples and explained them. Check out the set theory to be familiar with various concepts of Sets.
The Following Venn Diagrams show the Relationship between Sets. The Relation between them is explained with a description below the diagram.
Union of Sets using Venn Diagrams
Union of Sets using Venn Diagrams in different cases like Disjoint Sets, A ⊂ B or B ⊂ A, neither subset of A or B is explained in the below figures.
The Above Venn Diagram is A U B when neither A is a subset of B or B is a subset of A.
The above diagram shows A U B when A is a subset of B.
A U B when A and B are disjoint sets
Intersection of Sets using Venn Diagram
Intersection of Sets using Venn Diagrams in different cases like Disjoint Sets, A ⊂ B or B ⊂ A, neither subset of A or B is explained in the below figures.
A ∩ B when neither A ⊂ B nor B ⊂ A. The Intersection Part is highlighted with a color.
In the case of disjoint sets, there will be no common elements thus
A ∩ B = ϕ No shaded part.
Difference of Two Sets using Venn Diagrams
Learn about the difference of sets in various cases like A – B when neither A ⊂ B nor B ⊂ A, when A and B are disjoint Sets in the below modules.
The above diagram depicts the relationship between sets when neither A ⊂ B nor B ⊂ A.
In the case of disjoint Sets, A-B results in A simply. That is better understood by Venn Diagrams
Relationship between Three Sets using Venn Diagrams
The Above Diagram describes the Union Operation between Three Sets i.e. A U B U C.
Intersection Operation between Three Sets A, B, C is given above. The Shaded Region includes the elements that are in Sets A, B, C.
Let’s have a synopsis of the uses of Set Theory and its Applications in Problems. If at all A is a finite set then the number of elements in Set A is given by n(A).
In the Case of Relationship Between Sets using Venn Diagrams two cases arise. Let A and B be two finite sets
a) In Case if A and B are finite sets and disjoint i.e. no common elements then Formulas for Union is as such
n(A ∪ B) = n(A) + n(B)
b) If A and B are not disjoint then
(i) n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
(ii) n(A ∪ B) = n(A – B) + n(B – A) + n(A ∩ B)
(iii) n(A) = n(A – B) + n(A ∩ B)
(iv) n(B) = n(B – A) + n(A ∩ B)
Let A, B, C be three finite Sets then
n(A ∪ B ∪ C) = n[(A ∪ B) ∪ C]
= n(A ∪ B) + n(C) – n[(A ∪ B) ∩ C]
= [n(A) + n(B) – n(A ∩ B)] + n(C) – n [(A ∩ C) ∪ (B ∩ C)]
= n(A) + n(B) + n(C) – n(A ∩ B) – n(A ∩ C) – n(B ∩ C) + n(A ∩ B ∩ C)[We know, (A ∩ C) ∩ (B ∩ C) = A ∩ B ∩ C]
Therefore, n(A ∪B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)