Question

Three sets $A,B$, and $C$ are defined as,

$\begin{array}{cc}A& =\{x:x\text{is a whole number less than equal to 10}\}\\ & \\ B& =\{x:x\text{is a letter in the word "ARITHMETIC"}\}\\ & \\ C& =\{x:x\text{is a prime number less than 30}\}\end{array}$

With respect to the cardinality of these sets, choose the correct option.

• A. $n\left(A\right)\ne n\left(B\right)\ne n\left(C\right)$
• B. $n\left(A\right)=n\left(C\right)\ne n\left(B\right)$
• C. $n\left(B\right)=n\left(C\right)\ne n\left(A\right)$
• D. $n\left(A\right)=n\left(B\right)=n\left(C\right)$

Answer 1

The correct option is C $n\left(B\right)=n\left(C\right)\ne n\left(A\right)$

The given sets in the Roaster form can be written as,

$\begin{array}{cc}A& =\{0,1,2,3,4,5,6,7,8,9,10\}\\ & \\ B& =\{A,R,I,T,H,M,E,T,I,C\}\\ & \\ C& =\{2,3,5,7,11,13,17,19,23,29\}\end{array}$

The cardinality or cardinal number of a set is defined as the number of elements present in the set. It is denoted by $n\left(X\right)$, where $X$ be any set.

For the given sets, the number of elements in set $A$ is $11$, in $B$ the number of elements is $10$, and in $C$, it is $10$.

Therefore, their cardinality is given as,

$\begin{array}{c}n\left(A\right)=11\\ \\ n\left(B\right)=10\\ \\ n\left(C\right)=10\end{array}$

Hence, it can be concluded that $n\left(B\right)=n\left(C\right)\ne n\left(A\right)$.