Question

If $\xi$ = {$x$ : $x\le 10$, $xϵN$}, A = {$x$ : $x\ge 4$}, and B = {$x$ : $2<x<7$} ; find the number of equal sets.

Answer 1

According to the given sets, we have

$\xi =$ { $1,2,3,4,5,6,7,8,9,10$ }
$A=$ { $4,5,6,7,8,9,10$ }
$B=$ { $3,4,5,6$ }
Union of two sets will have all the elements of both the sets.
So, $A\cup B=$ { $3,4,5,6,7,8,9,10$ }
And $(A\cup B{)}^{\prime }$ will have the elements of $\xi$ which are not in $(A\cup B)$
So, $(A\cup B{)}^{\prime }=\xi -(A\cup B)=$ { $1,2$ } -- (1)
Now, $A^{\prime }=\xi -A=$ { $1,2,3$ }
And $B^{\prime }=\xi -A=$ { $1,2,7,8,9,10$ }
Intersection of two sets will have the elements common to both sets.
So, $A^{\prime }\cap B^{\prime }=$ { $1,2$ } -- (2)
From (1) and (2), we have
$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$
Also, $(A\cap B)=$ { $4,5,6$ }
And $(A\cap B{)}^{\prime }=\xi -(A\cap B)=$ { $1,2,3,7,8,9,10$ } -- (1)

$A^{\prime }\cup B^{\prime }=$ { $1,2,3,7,8,9,10$ } -- (2)
From (1) and (2), we have
$(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$