Question

If $$A\subset B,n\left(A\right)=5$$ and $$n\left(B\right)=7$$, then $$n(A\cup B)=$$_ _ _ _ _ _ _ _ _.
(a) 5
(b) 7
(c) 2
(d) 12

Answer 1

Hint: Use the formula for union of two sets that is $$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$$. Now, since $$A\subset B$$, therefore put $$n(A\cap B)=n\left(A\right)$$, then use the given information that is $$n\left(A\right)=5$$ and $$n\left(B\right)=7$$.

Here, we are given two sets A and B such that $$A\subset B$$, $$n\left(A\right)=5$$ and $$n\left(B\right)=7$$. We have to find the value of $$n(A\cup B)$$.
Before proceeding with the question, we must know some of the terminologies related to sets.
First of all, a ‘set’ is a collection of well-defined and distinct objects. The most basic property of a set is that it has elements. The number of elements of a set, say A is shown by $$n\left(A\right)$$.
Here, in the question we have $$n\left(A\right)=5$$ and $$n\left(B\right)=7$$, that means the number of elements in set A is 5, while the number of elements in set B is 7.
Now, a ‘subset’ is a set which is contained in another set. We can also put it as, if we have a set P which is a subset of another set Q, then P is contained in Q as all the elements of set P are elements of Q. This relationship is shown by $$P\subset Q$$.
We can show it diagrammatically as,

Here, $$P\subset Q$$, that means P is contained in Q as P is a subset of Q.
In question, we are given that $$A\subset B$$, that means that A is a subset of B or A is contained in B. We can show them as

Now, union of two sets say P and Q is the set of elements which are in P, in Q or both P and Q. For example, if P = {1, 3, 5, 7} and Q = {1, 2, 4, 6, 7}, then union of P and Q which is shown as $$P\cup Q=\{1,2,3,4,5,6,7\}$$
Diagrammatically, the shaded portion is $$P\cup Q$$ which is as follows

The formula for $$n(P\cup Q)=n\left(P\right)+n\left(Q\right)-n(P\cap Q)$$.
Here, $$P\cap Q$$ is the area common to both P and Q.
Now, in the given question, we have to find $$n(A\cup B)$$, that is, the number of elements in A union B.
We can show $$A\cup B$$ by a shaded portion which is as follows.

Here, $$A\subset B$$ and $$n\left(A\right)=5$$ and $$n\left(B\right)=7$$.
Here, we can see that the portion common to the set A and B that is $$(A\cap B)$$ is nothing but set A. Therefore, here we have $$n(A\cap B)=n\left(A\right)=5$$.
As we know that $$n(P\cup Q)=n\left(P\right)+n\left(Q\right)-n(P\cap Q)$$, therefore to get $$n(A\cup B)$$, we will put A and B in place of P and Q respectively, we will get
$$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$$
Since, we have found that $$n(A\cap B)=n\left(A\right)=5$$.
Therefore we get, $$n(A\cup B)=n\left(A\right)+n\left(B\right)-n\left(A\right)$$
By putting the values of n (A) and n (B), we get,
$$n(A\cup B)=5+7-5$$
$$n(A\cup B)=7$$
Therefore, we get $$n(A\cup B)=7$$
Hence, option (b) is correct.

Note: Students must note that whenever $$A\subset B$$, that is A is subset of B, then $$n(A\cup B)$$, that is the number of elements in A union B is equal to number of elements in set B that is, $$n(A\cup B)=n\left(B\right)$$ when $$A\subset B$$. Also, some students make this mistake of writing $$n(A\cup B)=n\left(A\right)+n\left(B\right)$$ which is wrong. They must remember to subtract $$n(A\cap B)$$ as well. Hence, $$n(A\cap B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$$.