Question

If \[n\left( A \right)=3\] , \[n(B)=6\] and \[A\subseteq B\] . Then number of elements in \[A\cup B\] is equal to
(A) 3
(B) 9
(C) 6
(D) None of these

Answer 1

Hint: This question deals with the concepts of set theory, where we are given the number of elements in set A, set B respectively. We have to calculate the number of elements in \[A\cup B\] which implies that either an element belongs to A or B or both. We are given that \[A\subseteq B\] which means that all elements that belong to A also belong to B.

Complete step-by-step answer:
A set is a well- defined collection of objects.
In this question it is given that number of elements in set A, \[n\left( A \right)=3\]
And number of elements in set B, \[n(B)=6\]
It is also given that \[A\subseteq B\] , which implies that A is subset of B . A subset means that all the elements of that set are in other that, that is, if \[A\subseteq B\] this implies that all the elements of A are in B but it is not necessary that all elements of B are in A also. Number of elements in B is greater than or equal to the number of elements in B.
In this question, we have the number of elements in B greater than the number of elements in A. So, this implies that B will have all the elements of A along with some other elements.
Number of elements in A = 3
Number of elements in B = 6
Number of elements in \[A\cup B\] implies that all the elements which are either in A or in B and we know that B contains all the elements of A,
It is given that \[A\subseteq B\] which means \[A\cup B=B\]
So, \[n\left( A\cup B \right)=n\left( B \right)\]
Which implies that,
Number of elements in \[A\cup B\] , \[n\left( A\cup B \right)=6\]
Hence, option (c) is the correct answer.
So, the correct answer is “Option C”.

Note: The main thing to keep in mind is the concept of subsets. We should remember the basic definition of subset that \[A\subseteq B\] means that all elements that belong to A also belong to B. We should also keep in mind that if \[A\subseteq B\] then, \[A\cup B=B\] . Keep in mind the basic formulas and definitions of set theory.