Find the relations between A, B and C when $[(A\cap B)\cup C]-A=(A\cap B)-C$

So we can write it as: $[(A\cup C)\cap(B\cup C)]-A=(A\cap B)-C$. Here comes the problem, though. Can I just assume that $[(A\cup C)\cap(B\cup C)]=(A\cap B)$? If I can, I can go on from there with: $C\subseteq A, C\subseteq B$ and eventually for the whole thing to hold $A=C$ which then shows that $A\subseteq B$. Can I do this that way, however? I feel that I'm assuming too much by simply comparing $[(A\cup C)\cap(B\cup C)]$ and $(A\cap B)$ to each other.


Let me name the two sets: $$ P=[(A\cap B)\cup C]-A\qquad Q=(A\cap B)-C $$ Your task is to determine the conditions on $A, B, C$ which will ensure that $P=Q$.

Helpful result 1: $P=C-A$.

A Venn diagram will make this clear, or you can show it by using set equivalences.

Helpful result 2: $P\cap Q = \varnothing$.

Again, use a Venn diagram or set equivalences. As a consequence we must conclude that both $P\text{ and }Q$ are empty, since the only way two sets can be equal but have no elements in common is for them both to be empty.

So we have two conditions: $$ \begin{align} P &= [(A\cap B)\cup C]-A=\varnothing\\ Q &= (A\cap B)-C=\varnothing \end{align} $$ Let's look at the first condition, $P=C-A=\varnothing$. This is true if and only if $C\subseteq A$.

Now apply this condition to $Q$. As above, $(A\cap B)-C=\varnothing$ if and only if $A\cap B\subseteq C$, so our condition is that

$P\text{ and }Q$ are equal if and only if $A\cap B\subseteq C\subseteq A$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.