Advanced Set Functions

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.

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

       X⋂Y=Y⋂X

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

(X⋂Y)⋂Z=X⋂(Y⋂Z)

3. Law of empty set: Let be an empty set then 

       ⋂X=φ

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

       U⋂X=X

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

i.e., X⋂X=X 

6. 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.