Question

here are two sets $A$ and $B$ such that $n(A - B) = 8 + 2x$ , $n(B - A) = 8 + 2x$ , $n(A \cap B) = x$ and $n(A) = n(B)$ . What is the value of $n(A \cap B)$?
A. 26
B. 50
C. 24
D. None of these

Answer 1

Hint: Use the properties of sets relating to the difference and the intersection of two sets. Considering there are two sets $A$ and $B$ in the universe, then $n(A) = n(A - B) + n(A \cap B)$ and $n(B) + n(B - A) + n(A \cap B)$ , where $ \cap $ represents the intersection of the two sets.

Formula used:
 $n(A) = n(A - B) + n(A \cap B)$ .

Complete step by step Solution:
It is given that $n(A - B) = 8 + 2x$ and $n(B - A) = 8 + 2x$ .
Now, using the properties of sets, we know that:
$n(A) = n(A - B) + n(A \cap B)$ .
It is also given to us that $n(A \cap B) = x$ .
Therefore, substituting the given values above, we get:
$n(A) = 8 + 2x + x$
On simplifying,
$n(A) = 8 + 3x$ … (1)
Similarly, using the properties of sets, we know that:
$n(B) = n(B - A) + n(A \cap B)$
Substituting all the given values, we get:
$n(B) = 6x + x$
On simplifying,
$n(B) = 7x$ … (2)
It is also given to us that $n(A) = n(B)$ .
Hence, using equations (1) and (2), we get:
$8 + 3x = 7x$
This, on simplifying further gives us $x = 2$ .
As $n(A \cap B) = x$ ,
Therefore, \[\;n(A \cap B) = 2\] .

Therefore, the correct option is D.

Note:Questions related to sets are much easier to solve using Venn Diagrams as they give us a very clear idea of the intersections, unions, differences, complements, and other properties of sets. This reduces the need to learn the formulas regarding their properties.