Ideas in Geometry/Instructive examples/9 from 2.1, Proof by Picture using Venn Diagrams
Prove that: (X-Y) u (Y-X)=(X u Y)-(X n Y)
First, look at each side of the equation separately. The first part of the left side is (X-Y). This means X complement Y, which is the set of all the elements that are in X and are not in Y. X will be shaded, but not where it overlaps Y. X-Y is shown in image 1 below.
The second half of the left side is Y-X. Y complement X is the set of all the elements that are in Y and not in X. Like the first half of the left side, Y will be shaded, but not where it overlaps with X. This is shown in image 2 below.
In between these two sets is a u, or union. This refers to all the elements in X-Y and Y-X. This means that everything in both pictures that we have constructed will combine and everything will be shaded. This is shown in image 3 below.
After we have finished with the left side of the equation, we can move on to the right side. The right side begins with (X u Y). As I stated before, union means that the set of all the elements in X and all the elements in Y. For this picture, we will shade both X and Y entirely. This is shown in image 4 below.
The second half of the right side of the equation is (X n Y). This means that given two sets X and Y, X intersect Y is the set of all the elements that are simultaneously in X and in Y. Pictorially, the only part that will be shaded is where X and Y overlap. For this, refer to image 5 below.
In between (X u Y) and (X n Y) on the right side is a complement. This means that the set of all the elements that are in X u Y and are not in X n Y will be shaded in. Look at image 6 below.
After we have finished working on both sides. We need to compare our final product. Looking at the pictures from each side of the equation (pictures 3 and 6), we can conclude that they are equal due to the fact that they are both shaded in the same areas.