Disprove using set identities

discrete mathematicselementary-set-theory

Let $A,B,C$ be sets. Draw a Venn diagram and color the region $(A-C)\cap(B- C)$.
disprove that $(A-C)\cap(B- C) = \emptyset$.

Already done the first part, but the proving part I am stuck at.
I have no clue on how to do this. Please show/guide me

Best Answer

"Already done the first part"

And what'd you get?

I get:

![enter image description here

Where $I$ and $IV$ make up $A-C$ and $II$ and $IV$ make up $B-C$ and $IV$ make up $(A-C) \cap (B-C)$ which need not be empty.

"but the proving part I am stuck at"

Can't prove what isn't true. So we need a counter example. Any normal set will do. But I already had those symbols in (for another reason) so I'll just use them.

Let $A = \{1,4,5,7\}$, Let $B=\{2,4,6,7\}$ and let $C = \{3,5,6,7\}$.

Then $A-C=\{1,4,5,7\}-\{3,5,6,7\} = \{1,4\}$ and $B-C=\{2,4,6,7\} -\{3,5,6,7\}=\{2,4\}$.

And $(A-C)\cap (B-C) = \{1,4\}\cap \{2,4\} = \{4\}$.

.....

Or if I wanted to be clever and as simple and blunt as possible.

Let $A=B=\{x\}$ and let $C=\emptyset$. Then

$$\begin{align} A-C&= A-\emptyset\\ &=A \\ &=B \\ &=\{x\} \end{align}$$ and

$$\begin{align} B-C&=B-\emptyset \\ &=B\\ &=A \\ &=\{x\} \end{align}$$ and

$$\begin{align} (A-C)\cap (B-C)&=A\cap B\\ & =A\cap A \\ &= A\\ &=B\\ &=\{x\} \end{align}$$

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