Solving Venn Diagrams

Venn diagrams are often a good way to illustrate the solution of set operations. For some, visualizing the solution is easy but for others, shading in the proper portion of a Venn diagram can be a challenge. Luckily, there’s a technique which will allow us to work with the sets first, without working about the diagram. To see how this technique works let’s first start with the following three circle Venn diagram.

Suppose we want to shade the set $(A \cup \bar{B}) \cap C$ . That’s what we need to shade in. Now, let’s number the regions of the Venn diagram. Be sure to number the region outside of the circles as well.

We see that set $ A = {1, 2, 3, 4} $ , $ B = {2, 3, 5, 6} $ and $ C = { 3, 4, 6, 7} $ . From here out, we’ll work with the numbers of each set and build up until we have $(A \cup \bar{B}) \cap C$ .

First, $\bar{B} = {1, 4, 7, 8}$ . (That’s all numbers not in B.)
$A \cup \bar{B} = {1, 2, 3, 4, 7, 8}$ . (All elements of set or $\bar{B}$ .
Finally, $(A \cup \bar{B}) \cap C = {3, 4, 7}$ . (All elements of $A \cup \bar{B}$ and , since we’re dealing with the intersection of these sets.)