Question

Which among the following is a singleton set?

• A. $M=\left\{5\right\}$
• B. $N=\{k:k\le 1,k\in N\}$
• C. $P=\{k:k\text{is a vowel in the word "SETS"}\}$
• D. $Q=\{k:k\text{is the solution of}{x}^3-3x^2+3x-1=0\}$

Answer 1

Answer: (A), (B), (C), (D)

$Q=\{k:k\text{is the solution of}{x}^3-3x^2+3x-1=0\}$

A singleton set is a set with just one element.
Here, Let's write the following in the roster form and find their cardinality.
$M=\left\{5\right\}\Rightarrow n\left(M\right)=1$

$N=\{k:k\le 1,k\in N\}\Rightarrow N=\left\{1\right\}$
$\Rightarrow n\left(N\right)=1$

$P=\{k:k\text{is a vowel in the word "SETS"}\}$
$\Rightarrow P=\left\{E\right\}\Rightarrow n\left(P\right)=1$

$Q=\{k:k\text{is the solution of}{x}^3-3x^2+3x-1=0\}$
Now, $x^3-3x^2+3x-1=0$
$\Rightarrow (x-1{)}^3=0$
Hence, there are three identical solutions of the equation.
$\Rightarrow Q=\left\{1\right\}\Rightarrow n\left(Q\right)=1$