Question

State which of the following sets are finite or infinite:
$\left(i\right)$ $\{x:x\in N\text{and}(x-1\left)\right(x-2)=0\}$
$\left(ii\right)$ $\{x:x\in N\text{and}{x}^2=4\}$
$\left(iii\right)$ $\{x:x\in N\text{and}2x-1=0\}$
$\left(iv\right)$ $\{x:x\in N\text{and}x\text{is prime}\}$
$\left(v\right)$ $\{x:x\in N\text{and}x\text{is odd}\}$

Answer 1

$\left(i\right)$Since $(x-1)(x-2)=0$
$\therefore x=1,2$
So, given set $=\{1,2\}$
The set has 2 elemnets (countable).
Hence, it is finite

$\left(ii\right)$Given : $x^2=4$
$\Rightarrow (x-2)(x+2)=0$
$\Rightarrow x=\pm 2$
But $x\in N$, it cannot be a negative number.
So, given set $=\left\{2\right\}$
Since the number of elemnts of set is countable.
$\therefore$ It is finite.

$\left(iii\right)$Given, $2x-1=0$
$\Rightarrow x=\frac{1}{2}$
But $x\in N$, it cannot be a rational number.
Thus, given set has no elements.
$\therefore$ Given set $=\varphi$ (null set)
So, given set is fnite

$\left(iv\right)$Given, $x$ is prime and $x\in N$
So, $x=2,3,5,7,11,13,17,...$
There are infinite number of prime numbers.
So, this is an infinite set.

$\left(v\right)$Given, $x$ is odd and $x\in N$
So, $x=1,3,5,7,9,...$
There are infinite number of odd numbers.
So, this is an infinite set.