The number of elements of the set $\left\{x:x\in N,x^2=1\right\}$ , where N is the set of all natural numbers is

Put $x=1,1^2=1\Rightarrow 1\in N$

$x=2,2^2\ne 1\Rightarrow 2\in N$  but not satisfied

So we observe that the above condition satisfied for $x=1$

So, the set contain only one element