# Union of Sets – Definition and Venn Diagram with Examples

Here you will learn what is the union of sets with definition and venn diagram representation and examples.

Let’s begin –

## What is the Union of Sets ?

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

We shall use the notation $$A \cup B$$  (read as “A union B”) to denote the union of A and B. Thus, $$A \cup B$$ = {x : x $$\in$$ A or x $$\in$$ B}.

Clearly,  x $$\in$$ $$A \cup B$$  $$\iff$$  x $$\in$$ A   or   x $$\in$$ B.

And,  x $$\notin$$ $$A \cup B$$  $$\iff$$  x $$\notin$$ A   or   x $$\notin$$ B.

In the given figure whole shaded part represents $$A \cup B$$. This is the venn diagram for union of sets.

It is evident from the definition the A $$\subseteq$$ $$A \cup B$$, B$$\subseteq$$ $$A \cup B$$.

If A and B are two sets such that A $$\subset$$ B, then $$A \cup B$$ = B. Also, $$A \cup B$$ = A, if B$$\subset$$ A.

Example : If A = {1, 2, 3} and B = {1, 3, 5, 7}, then $$A \cup B$$ = {1, 2, 3, 5, 7}.

Example : If A = {1, 2, 3}, B = {3, 5} and C = {4, 7, 8}. Then $$A \cup B \cup C$$ = {1, 2, 3, 4, 5, 7, 8}.

## Formula to Find Number of Elements in A Union B

If A, B and C are finite sets, and U be the finite universal set, then

n($$A \cup B$$) = n(A) + n(B) – n($$A \cap B$$)

where, n(A) = number of elements in set A

n(B) = number of elements in set B

n($$A \cap B$$) = number of elements in intersection of sets A and B

Also Read : Other Formulas and Operation of Sets

Example : If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n($$X \cap Y$$) = 2, then find n($$X \cup Y$$).

Solution : By using the above formula,

n($$X \cup Y$$) = n(X) + n(Y) – n($$X \cap Y$$)

$$\implies$$  n($$X \cup Y$$) = 17 + 23 – 2

$$\implies$$  n($$X \cup Y$$) = 38