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