A union of well-arranged objects is known as a set. For example, a combination of real numbers is known as a set of real numbers. Each object in a set is known as an element of the set.

A set is denoted by capital alphabet, i.e., A,B,C,…P,Q,R,…X,Y,Z,…. Elements in a set are denoted between brackets, i.e., {,,,,,,}

If a set has no elements, it is known as an empty set or null set. Empty set is denoted by {} or

Function: A function is an equation that generates a set of ordered pairs. A set of ordered pairs is defined as {input, output}. For each input, there should be one output

If an ordered pair is collected in such a way that one object is taken from each set, then it is known as a relation. For example, if an object x is taken from one set and another object y is taken from a second set, then (x,y) is the ordered pair in the relation

Subset of a set: If all elements of a set X are elements of another set Y, then X is known as the subset of set Y. For example, X={2,4,6,8,10} and Y={1,2,3,4,5,6,7,8,9,10,11}, then X is known as a subset of Y. Subset is denoted as X⊂Y

Intersection of sets: Suppose two sets A and B are subsets of universal set U; if some elements of A and B are common, then the intersection of both sets is A∩B

## Intersection operation in sets

Let X and Y be two sets. These sets are subsets of universal set U, i.e., X⊂U and YU.

Let x,y be elements of set A, i.e.,x,y∈A

And x,z be elements of set B, i.e., x,z∈B

Then the intersection of set A and B is given by A⋂B={x,y}⋂x,z or A⋂B={x} as x is a common element between both sets A and B.

## Properties of intersection of sets

Intersection of sets has some properties such as commutative, associative, distributive, idempotent laws and laws of universal set and empty set.

Commutative law: If X and Y are two sets, then according to the commutative law

X⋂Y=Y⋂X

Associative law: If X, Y and Z are three sets, then according to the associative law,

(X⋂Y)⋂Z=X⋂(Y⋂Z)

Law of empty set: Let be an empty set then

⋂X=φ

Law of universal set: Let U be a universal set then

U⋂X=X

Idempotent law: Intersection of a set with the same set results in a set itself,

i.e., X⋂X=X

Distributive law: If X, Y and Z are three sets, then according to the distributive law,

X⋂(Y∪Z)=(X⋂Y)∪(X⋂Z)

## What are different set operations?

A group of objects is referred to as a set. Each object in a set is referred to as an ‘Element.’ There are three ways to represent a set: statement form, roster form and set-builder form. Set operations are actions performed on two or more sets to create a relationship between them. The operations on sets are divided into four categories:

Union of sets

Intersection of sets

Complement of a set

Difference between sets/relative Complement

## Union and intersection of sets

Union of sets: Combination of elements of two sets is known as union of sets. Let P and Q be two sets, then the union of P and Q is defined as P∪Q. Let P={a,b} and Q={c,d}, then P∪Q=a,bc,d={a,b,c,d}.

Intersection of sets: Taking common elements between two sets is known as the intersection of sets. Let P and Q be two sets, then the union of P and Q is defined as PQ. Let P={a,b} and Q={c,d}, then P∪Q=a,bb,d={b}.

### Conclusion

A set is a collection of well defined objects.A set is represented in curly brackets in which each element is separated by a comma.A set can be empty also which is denoted by . We can apply different operations on sets like union,intersection,complement and difference.We also discussed what is intersection of a set and some of its properties.