Question

If $\xi$ = {$x$ : $x\le 10$, $xϵN$}, A = {$x$ : $x\ge 4$}, and B = {$x$ : $2<x<7$} ; find number of elements in $A^{{}^{\prime }}\cup B^{{}^{\prime }}$

Answer 1

$\xi =$ { $1,2,3,4,5,6,7,8,9,10$ } as $x$ is a natural number less than or equal to $10$.
$A=$ { $4,5,6,7,8,9,10$ } as $x$ is a number greater than or equal to $4$ and the universal set is already defined as $\xi$.
And, $B=$ { $3,4,5,6$ } as $x$ is a natural number less than $7$ and more than $2$.
And ${\left(A\right)}^{{}^{\prime }}=\xi -A=$ {$1,2,3$ } (Since, $\xi -A$ will have the elements of $\xi$ which are not in $A$ ).
Similarly, ${\left(B\right)}^{{}^{\prime }}=\xi -B=$ {$1,2,7,8,9,10$ }
Union of two sets has the elements of both the sets.
So, ${\left(A\right)}^{{}^{\prime }}\cup {\left(B\right)}^{{}^{\prime }}=$ { $1,2,3,7,8,9,10$ }.