## Union of sets:

The basic operations that can be performed on sets are as follows:

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In mathematics, we perform certain operations like addition, subtraction, multiplication, etc. These operators generally take two or more operands and give a result based on the operation performed. Similarly, in set theory usually certain operations are performed on two or more sets to get a new set of elements based on the operation performed.

In the upcoming discussions we will study about the union thoroughly. Let us consider a universal set U such that A and B are the subsets of this universal set. The union of two sets A and B is defined as the set of all the elements which lie in set A and set B or both the elements in A and B altogether. The union of set is denoted by the symbol ‘∪’.

In the given Venn diagram, the red colored portion represents the union of both the sets A and B.

Thus, the union of two sets A and B is given by a set C, which is also a subset of the universal set U such that C consists of all those elements or members which are either in set A or set B or in both A and B i.e.,

A∪B={x:x ∈A or x∈B}

Let us go through an example to make it clearer.

**Example: Let U be a universal set consisting of all the natural numbers less than 20 and set A and B be a subset of U defined as A={2,5,9,15,19} and B={8,9,10,13,15 ,17}. Find A∪B.**

**Solution:** Given U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}

A={2,5,9,15,19}

B={8,9,10,13,15 ,17}

A∪B={2,5,8,9,10,13,15,17,19}

This can be represented using the following Venn diagram:

**Properties of Union of two Sets:**

**i) Commutative Law:** The union of two or more sets follows the commutative law i.e., if we have two sets A and B then,

A∪B=B∪A

**ii) Associative Law:** The union operation follows the associative law i.e., if we have three sets A, B and C then

(A∪B)∪C = A∪(B∪C)

**iii) Identity Law:** The union of an empty set with any set A gives the set itself i.e.,

A∪∅=A

**iv) Idempotent Law:** The union of any set A with itself gives the set A i.e.,

A∪A=A

**v) Domination Law:** The union of a universal set U with its subset A gives the universal set itself.

A∪U=U

This is all about the union operation. Quench your thirst for knowledge with us at BYJU’s. To learn about intersection of sets, download BYJU’S-the learning app.

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