Question

For any three sets $X,Y\&Z,X\cap (Y\cap Z)$ is equal to

• A. $(X\cap Y)\cap Z$
• B. $(Y\cap X)\cap Z$
• C. $Y\cap (X\cap Z)$
• D. $Y\cap (Z\cap X)$

Answer 1

Answer: (A), (B), (C), (D)

$Y\cap (Z\cap X)$

Given three sets $X,Y\&Z$
Now, from associative property of intersection of sets:
$X\cap (Y\cap Z)=(X\cap Y)\cap Z$
Here, applying the commutative property of intersection of sets:
$(X\cap Y)\cap Z=(Y\cap X)\cap Z$
Now, applying the associative property of intersection of sets:
$(Y\cap X)\cap Z=Y\cap (X\cap Z)$
Applying the commutative property of intersection of sets:
$Y\cap (X\cap Z)=Y\cap (Z\cap X).$