Question

Statement I The set of letters needed to spell 'CATARACT' and the set of letters needed to spell 'TRACT' are equal.
Statement II All the subsets of the set $\{-1,0,1\}$ are $\varphi ,\{-1\},\{0\},\{1\},\{-1,0\},\{-1,1\}$ and $\{0,1\}$

• A. Statement I is true
• B. Statement II is true
• C. Both are true
• D. Both are false

Answer 1

Answer: (A)

Let $X$ be the set of letters in word 'CATARACT'. Then,
$X=\{C,A,T,R\}$
Let $Y$ be the set of letters in word 'TRACT'. Then,
$Y=\{T,R,A,C,T\}=\{T,R,A,C\}$
Since, every element in $X$ is in $Y$ and every element in $Y$ is in $X$. It follows that $X=Y$.
Hence, Statement I is true.
Let $A=\{-1,0,1\}$. The subset of $A$ having no element is the empty set $\varphi$. The subsets of $A$ having one element are $\{-1\}$, $\{0\}$ and $\{1\}$. The subsets of $A$ having two elements are $\{-1,0\},\{-1,1\}$ and $\{0,1\}$. The subset of $A$ having three elements of $A$ is $A$ itself. So, all the subsets of $A$ are $\varphi ,\{-1\}$, $\{0\},\{1\},\{-1,0\},\{-1,1\},\{0,1\}$ and $\{-1,0,1\}$.
Hence, Statement II is false.