Question

Enter $1$ if the below relation holds true and $0$ if the above relation doesnot hold true:

A $\cup$ (B${}^{{}^{\prime }}$$\cup$ A${}^{{}^{\prime }}$) = A $\cup$ (B $\cap$ A)${}^{{}^{\prime }}$

Answer 1

$\xi =$ { $11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30$ } as $x$ is a natural number more than $10$ and less than equal to $30$
$A=$ { $11,12,13,14,15,16,17,18,19$ } as $x$ is a number greater than or equal to $20$ with the limit of the universal set already defined as $\xi$
And, $B=$ { $16,17,18,19,20,21,22,23,24$ } as $x$ is a natural number less than $25$ and more than $15$
For LHS:
${\left(A\right)}^{{}^{\prime }}=\xi -A=$ {$20,21,22,23,24,25,26,27,28,29,30$ } (Since, $\xi -A$ will have the elements of $\xi$ which are not in $A$ )
Similarly, ${\left(B\right)}^{{}^{\prime }}=\xi -B=$ {$11,12,13,14,15,25,26,27,28,29,30$ }
Union of two sets has the elements of both the sets.
So, $B^{{}^{\prime }}\cup A^{{}^{\prime }}=$ { $11,12,13,14,15,20,21,22,23,24,25,26,27,28,29,30$ }
And $A\cup (B^{{}^{\prime }}\cup A^{{}^{\prime }})=$ {$11,12,13,14,15,16,17,18,1920,21,22,23,24,25,26,27,28,29,30$ } - (1)
For RHS:
Intersection of two sets has the elements common in both the sets.
$=>B\cap A=$ {$16,17,18,19$ }
$=>{(B\cap A)}^{{}^{\prime }}=\xi -(B\cap A)=$ {$11,12,13,14,15,20,21,22,23,24,25,26,27,28,29,30$ } (Since, $\xi -(B\cap A)$ will have the elements of $\xi$ which are not in $(B\cap A)$ )

So, $A\cup {(B\cap A)}^{{}^{\prime }}=$ { $11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30$ } -- (2)
From 1, 2
$A\cup (B^{{}^{\prime }}\cup A^{{}^{\prime }})=A\cup {(B\cap A)}^{{}^{\prime }}$