Question

In the following, state whether $A = B$ or not:
(i) $A = \left\{ {a,b,c,d} \right\}$ $B = \left\{ {d,c,b,a} \right\}$
(ii) $A = \left\{ {4,8,12,16} \right\}$ $B = \left\{ {8,4,16,18} \right\}$
(iii) $A = \left\{ {2,4,6,8,10} \right\}$ $B = \left\{ {x:x{\text{ is a positive even integer and }}x \leqslant 10} \right\}$
(iv) $A = \left\{ {x:x{\text{ is a multiple of }}10} \right\}$ $B = \left\{ {10,15,20,25,30,...} \right\}$

Answer 1

Hint: Equality of two sets can be defined when each of the elements is the same in both the sets and sets have an equal number of elements. Use this information to check for the similarity of elements present in both A and B in each part. Write the sets in the roaster form where it is not.

Complete step-by-step answer:
Here in this question, we are given four pairs of set A and B and we need to find whether the given two sets in each part are equal or not equal.
Before starting with the solution let’s understand the equality of two sets. Two sets are called equal when both of them consist of the same elements and the arrangement of these elements is not of importance.
Considering part (i), here we have the first set as $A = \left\{ {a,b,c,d} \right\}$ and the second set as $B = \left\{ {d,c,b,a} \right\}$.
Here both of these sets have four elements and these elements are the same for both, i.e. $a,b,c,d \in A{\text{ and }}a,b,c,d \in B$ .
Therefore, we can conclude $A = B$ for part (i)
Now for part (ii), set A is given as $\left\{ {4,8,12,16} \right\}$ , and set B is given as $\left\{ {8,4,16,18} \right\}$.
Here both sets have four elements, where $4,8{\text{ and }}16$ is in both the sets. But $12{\text{ and }}18$ are the elements which are different in both A and B.
Thus, we can conclude $A \ne B$ for part (ii)
For part (iii), we have set A as $\left\{ {2,4,6,8,10} \right\}$ and set B as a set of positive even integers less than and equal to $10$ .
So the set B can be written as $B = \left\{ {2,4,6,8,10} \right\}$ .
Here both sets A and B have five elements and all the five elements are the same.
Therefore, we can say that $A = B$ for part (iii)
For part (iv), we have set A as a set of multiples of $10$ and set B is given as $B = \left\{ {10,15,20,25,30,...} \right\}$
So, the set A can be written as $A = \left\{ {10,20,30,40,50,...} \right\}$
Here both are infinite sets but in set B, numbers which are multiple of $5$ such as $15,25....$ which can never be present in a set of multiples of ten.
Thus, we can say $A \ne B$ for part (iv).


Note: Remember the use of roaster form of representing sets is the most important part of questions like this. The two sets having a dissimilar number of elements can never be equal. This can be used as an easy trick to eliminate the sets with a different number of elements.