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.union of sets

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

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