Question

If X, Y, Z are sets, prove that
(X ∩ Y ) ∪ (X’ ∩ Z) ∪ (Y ∩ Z) = (X ∩ Y) ∪ (X’∩ Z)

Answer 1

looks fun but i have to eat i will be back and answer this if someone else doesn't

Answer 2

Two show two sets are equal A=B.
We need to show A is a subset of B by showing every element of A is in B.
We need to show B is a subset of A by showing every element of B is in A.

So let's do this first part
Suppose
\[x \in (X \cap Y) \cup (X' \cap Z) \cup (Y \cap Z)\]
we want to show that
\[x \in (X \cap Y) \cup (X' \cap Z)\]
if we do this then we can show that \[A \subseteq B\]

Answer 3

We have
\[x \in X \cap Y\]
or
\[x \in X' \cap Z\]
or
\[x \in Y \cap Z\]
we have three cases

Answer 4

Show

\[Y\cap Z\subseteq(X\cap Y)\cup(X'\cap Z)\]

Answer 5

right

Answer 6

thanks much