Question

Which of the following is a singleton set?

• A. $\{x:x\text{is a point where a tangent touches a circle}\}$
• B. $\{y:y\text{is a prime number},20\le y<23\}$
• C. $\{a:a\in N,a^2+5a+6=0\}$
• D. $\{b:b\in N,3b^2-5b-2=0\}$
• E. $\text{Set of all people born in May 2000}$

Answer 1

Answer: (A), (D)

$\{b:b\in N,3b^2-5b-2=0\}$

A singleton set is a set with cardianality 1.

Let's take a look at the examples one by one.

$A.\{x:x\text{is a point where a tangent touches a circle}\}$
Here, there is only one tangent touching a circle from a point lying on the circle.
$\Rightarrow$ This is a singleton set.

$B.\{y:y\text{is a prime number},20\le y<23\}$
$y\in N\text{and}20\le y<23$
$\Rightarrow y=20,21,22$
And each of the following is a composite number
$\Rightarrow$ This is an empty set.

$C.\{a:a\in N,a^2+5a+6=0\}$
$a^2+5a+6=0$
$\Rightarrow a^2+2a+3a+6=0$
$\Rightarrow (a+2)(a+3)=0$
$\Rightarrow a=-2\text{or}a=-3$
Since, both are not natural numbers, this means that the set is an empty set.

$D.\{b:b\in N,3b^2-5b-2=0\}$
$3b^2-5b-2=0$
$\Rightarrow 3b^2-6b+b-2=0$
$\Rightarrow (3b+1)(b-2)=0$
$\Rightarrow b=-\frac{1}{3}\text{or}b=2$
$\because b\in N\Rightarrow b=2$
$\Rightarrow$ The set is a singleton set.

$E.$ Set of all people born in May 2000.
Here, The numbers of people born in May 2000 is definitely greater than 1.
$\Rightarrow$ The set is neither singleton nor empty.