Maths Questions

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

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

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

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

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.

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.

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.