# Shading A Venn Diagram Using A Specific Equation

The expression is: $A\triangle(B\cap C')$. The $\triangle$ refers to $-$ and the $\cap$ refers to an intersection, whilst the $\;{}'$ refers to the prime of $C$.

There are no numbers or items included in this problem. I have drawn a Venn diagram with a background (for the universal set) and circles $A$, $B$ and $C$ which equally overlap. The problem is, I am unsure how to shade it. I also must provide the logic used to solve the equation. Thanks a bunch!

## 2 Answers

It might help to do this in steps, from the inside out, try drawing tree diagrams:

• in the first, simply shade C', that is all the area out side of $C$.
• next, you'd like to draw $B\cap C'$ i.e. the intersection of $B$ with the shaded area of the previews diagram, more simply it would be the area inside $B$ that was shaded in the first diagram.
• finally, in the third diagram, you need to shade the area inside A that was not shaded in the second diagram.

put into words the expression means, everything that is in A with the exception of($\Delta$) things that are in B and($\cap$) not($′$) in $C$.

At first this might be confusing, and you might need to draw the steps, but with practice you will learn to do it in your head, Venn diagrams are a very powerful tool for intuitively understanding logic.

• This answer was very confusing, but gave me an idea how to solve it. It would have been simpler had I known that A△(B∩C′) is equivalent to A∩(B∩C')' Thanks! – Shayna May 21 '13 at 8:32

A△(B∩C′) = A∩(B∩C')' They are equivalent! So, shade all of A that is not in B. Also shade the intersection of A, B & C (You should have shaded A, A/C AND A/B/C.).