Union of Sets using Venn Diagram

Learn how to represent the union of sets using Venn diagram. The union set operations can be visualized from the diagrammatic representation of sets.

The rectangular region represents the universal set U and the circular regions the subsets A and B. The shaded portion represents the set name below the diagram.

Let A and B be the two sets. The union of A and B is the set of all those elements which belong either to A or to B or both A and B.

Now we will use the notation A U B (which is read as ‘A union B’) to denote the union of set A and set B.

Thus, A U B = {x : x A or x B}.

Clearly, x A U B   

⇒ x A or x B

Similarly, if x ∉ A U B  

⇒ x ∉ A or x B

Therefore, the shaded portion in the adjoining figure represents A U B.

Union of Sets using Venn Diagram

Thus, we conclude from the definition of union of sets that A ⊆ A U B, B ⊆ A U B.


From the above Venn diagram the following theorems are obvious:

(i) A ∪ A = A                        (Idempotent theorem)

(ii) A ⋃ U = U                       (Theorem of ⋃) U is the universal set.

(iii) If A ⊆ B, then A ⋃ B = B

(iv) A ∪ B = B ∪ A                (Commutative theorem)

(v) A ∪ ϕ = A                      (Theorem of identity element, is the identity of ∪) 

(vi) A ⋃ A' = U                     (Theorem of ⋃) U is the universal set.


Notes:

A ∪ ϕ = ϕ ∪ A = A i.e. union of any set with the empty set is always the set itself.


Solved examples of union of sets using Venn diagram:

1. If A = {2, 5, 7} and B = {1, 2, 5, 8}. Find A U B using venn diagram.

Solution:

According to the given question we know, A = {2, 5, 7} and B = {1, 2, 5, 8}

Now let’s draw the venn diagram to find A union B.

Union using Venn Diagram

Therefore, from the Venn diagram we get A U B = {1, 2, 5, 7, 8}


2. From the adjoining figure find A union B.

Find A union B

Solution:

According to the adjoining figure we get;

Set A = {0, 1, 3, 5, 8}

Set B = {2, 5, 8, 9}

Therefore, A union B is the set of elements which in set A or in set B or in both.

Thus, A U B = {0, 1, 2, 3, 5, 8, 9}

Set Theory

Sets Theory

Representation of a Set

Types of Sets

Finite Sets and Infinite Sets

Power Set

Problems on Union of Sets

Problems on Intersection of Sets

Difference of two Sets

Complement of a Set

Problems on Complement of a Set

Problems on Operation on Sets

Word Problems on Sets

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Union of Sets using Venn Diagram

Intersection of Sets using Venn Diagram

Disjoint of Sets using Venn Diagram

Difference of Sets using Venn Diagram

Examples on Venn Diagram



8th Grade Math Practice

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