Ideas in Geometry/Instructive examples/Proving Sets Equal, Section 2.1

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In order to prove that X∩(Y∪Z) = (X∩Y)∪(X∩Z), you must first learn what the symbols mean. When there is an intersection symbol, ∩, it means that you want to shade in the common area between the two areas intersecting. When there is a union symbol, ∪, it means that you want to shade in everything included in both areas. To prove that X∩(Y∪Z) = (X∩Y)∪(X∩Z), it is best to break it into parts. As shown in the picture above, for X∩(Y∪Z) I first shaded in X. Then I shaded in Y∪Z. After doing those parts separately, I shaded where they had common shaded areas because of the intersection symbol. I did the same thing for (X∩Y)∪(X∩Z) by first shading in X∩Y. Then I shaded in X∩Z. Since there is a union symbol for this one, I then combined the shaded areas for both parts. After breaking up both sides in order to find the shaded region, you can see that the two ultimately equal each other.

By: Katy Anderson