# 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.

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.

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

2. From the adjoining figure 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