# Sets Questions

## What is the Identity Relation with Examples ?

Solution : Let A be a set. Then, the relation $$I_A$$ = {(a, a) : a $$\in$$ A} on A is called the identity relation on A. In other words, then the relation $$I_A$$ on A is called the identity relation if every element of A is related to itself only. Example : If A …

## What are Universal Relation with Example ?

Solution : Let A be a set. Then, A $$\times$$ A $$\subseteq$$ A $$\times$$ A and so it is a relation on A. This relation is called the universal relation on A. In other words, a relation R on a set is called universal relation, if each element of A is related to every element …

## What is Void or Empty Relation with Example ?

Solution : Let A be a set. Then, $$\phi$$ $$\subseteq$$ A $$\times$$ A and so it is a relation on A. This relation is called the void or empty relation on set A. In other words, a relation R on a set A is called void or empty relation, if no element of A is …

## Prove that the total number of subsets of a finite set containing n elements is $$2^n$$.

Solution : Let A be a finite set containing n elements. Let 0 $$\le$$ r $$\le$$ n. Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is $$^nC_r$$. Therefore, the number of subsets of A having …

## Prove that Empty Set is a Subset of Every Set.

Solution : Let A be any set and $$\phi$$ be the empty set. In order to show that $$\phi$$ $$\subseteq$$ A, we must show that every element of $$\phi$$ is an element of A also. But, $$\phi$$ contains no element. So, every element of $$\phi$$ is in A. Hence, $$\phi$$ is the subset of A.

## What is Singleton Set ?

Solution : A set consisting of a single element is called a singleton set. Example : The set {5} is a singleton set. Example : The set {x : x $$\in$$ N and $$x^2$$ = 9} is a singleton set equal to {3}. Note : The cardinal number of a singleton set is 1.

## What is Cardinality of Set ?

Solution : The cardinality of a set is the number of elements in a set. For example : Let A be a set : A = {1, 2, 4, 6} Set A contains 4 elements. Therefore, Cardinality of set is 4.

## Let A and B be two sets containing 2 elements and 4 elements, respectively. The number of subsets A$$\times$$B having 3 or more elements is

Solution : Given, n(A) = 2 and n(B) = 4 $$\therefore$$ n(A$$\times$$B) = 8 The number of subsets of (A$$times$$B) having 3 or more elements = $$^8C_3 + {^8C_4} + ….. + {^8C_8}$$ = $$2^8 – {^8C_0} – {^8C_1} – {^8C_2}$$ = 256 – 1 – 8 – 28 = 219     [$$\because$$ $$2^n$$ = …

## If A = {x,y}, then the power set of A is

Solution : Clearly P(A) = Power set of A = set of all subsets of A = {$$\phi$$, {x}, {y}, {x,y}} Similar Questions Let A and B be two sets containing 2 elements and 4 elements, respectively. The number of subsets A$$\times$$B having 3 or more elements is If aN = {ax : x $$\in$$ …

## If aN = {ax : x $$\in$$ N}, then the set 6N $$\cap$$ 8N is equal to

Solution : 6N = {6, 12, 18, 24, 30, …..} 8N = {8, 16, 24, 32, ….} $$\therefore$$ 6N $$\cap$$ 8N = {24, 48, …..} = 24N Similar Questions If A = {x,y}, then the power set of A is If A = {2, 4} and B = {3, 4, 5} then (A $$\cap$$ B) …