Question

Which of the following sets are equivalent sets?

• A. $\begin{array}{cc}A& =\{x:x\text{is a whole number less than equal to 7}\}\\ & \\ B& =\{x:x\text{is a letter in the word "MAXIMUM"}\}\end{array}$
• B. $\begin{array}{cc}A& =\{2,3,5,7,11\}\\ & \\ B& =\{x:x\text{is prime number less than 15}\}\end{array}$
• C. $\begin{array}{cc}A& =\{x:x\text{is a letter in the word "BLOOM"}\}\\ & \\ B& =\{x:x\text{is a letter in the word "BLOSSOM"}\}\end{array}$
• D. $\begin{array}{cc}A& =\{x:x\text{is a natural number less than equal to 4}\}\\ & \\ B& =\{x:x\text{is a letter in the word "NAME"}\}\end{array}$

Answer 1

The correct option is D $\begin{array}{cc}A& =\{x:x\text{is a natural number less than equal to 4}\}\\ & \\ B& =\{x:x\text{is a letter in the word "NAME"}\}\end{array}$

Two sets $X$ and $Y$ are said to be equivalent sets if they have the same number of elements.

For example, if $X=\{2,3,5\}$ and $Y=\{\text{Monday, Tuesday, Wednesday}}$, then they are called equivalent sets as the number of elements in the both the sets is same.

Now, consider $\begin{array}{cc}A& =\{x:x\text{is a whole number less than equal to 7}\}\\ & \\ B& =\{x:x\text{is a letter in the word "MAXIMUM"}\}\end{array}$

In roaster form, the sets can be represented as

$\begin{array}{cc}A& =\{0,1,2,3,4,5,6,7\}\\ & \\ B& =\{M,A,X,I,M,U,M\}\end{array}$

Since the number of elements are not same in both the sets, therefore they are not equivalent sets. Rather they are disjoint sets as they have no common element.

Consider, $\begin{array}{cc}A& =\{2,3,5,7,11\}\\ & \\ B& =\{x:x\text{is prime number less than 15}\}\end{array}$
The sets can be represented as

$\begin{array}{cc}A& =\{2,3,5,7,11\}\\ & \\ B& =\{2,3,5,7,11,13\}\end{array}$

These sets are also not equivalent as the number of elements are not same. Howerve, they are joint sets because they have some elements in common.

Consider, $\begin{array}{cc}A& =\{x:x\text{is a letter in the word "BLOOM"}\}\\ & \\ B& =\{x:x\text{is a letter in the word "BLOSSOM"}\}\end{array}$
The sets can be represented as,

$\begin{array}{cc}A& =\{B,L,O,O,M\}\\ & \\ B& =\{B,L,O,S,S,O,M\}\end{array}$

As the number of elements are not same, thus they are not equivalent sets. However, they have some common elements, therefore they are known as joint sets.

Consider $\begin{array}{cc}A& =\{x:x\text{is a natural number less than equal to 4}\}\\ & \\ B& =\{x:x\text{is a letter in the word "NAME"}\}\end{array}$
The sets can be represented as,

$\begin{array}{cc}A& =\{1,2,3,4\}\\ & \\ B& =\{N,A,M,E\}\end{array}$

Since the number of elements in both the sets are same, therefore they are known as equivalent sets.

Hence, out of the given sets, the sets
$\begin{array}{cc}A& =\{x:x\text{is a natural number less than equal to 4}\}\\ & \\ B& =\{x:x\text{is a letter in the word "NAME"}\}\end{array}$
are equivalent sets.