Ideas in Geometry/Instructive examples/Section 2.1

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Prove that: X - (X - Y) = The intersection of X and Y

This is saying, prove that subtracting the elements in X that are also in Y, from X as a whole is equivalent to the intersection of X and Y. Review: The intersection means the elements of X and Y, the elements the two sets have in common. Also, subtracting a set from a set, means excluding the elements the sets have in common. We can prove this using both sets of numbers or letters, and venn diagrams.

Proof One: Using Sets of Numbers Let's say set X={1,2,3,4}, and set Y={2,3,4,5} Deal with the left side first, and begin, as always, with what is in the parenthesis. (X-Y) So, we exclude all of the elements of Y from X. The elements they have in common are 2,3,4. We are left with X={1}

Now, we take the original set of X and subtract what we found for (X-Y) X - (X-Y) the same as {1,2,3,4} - {1}

We are left with {2,3,4} for the left side of the equation.

Now, let's deal with the right side, the intersection of X and Y. We know, based on the definition of intersection, this means the elements that are in both sets X and Y.

The elements that the sets have in common are 2,3,4. So, the right side of the equation is equal to {2,3,4} Thus, {2,3,4} = {2,3,4}, which proves X - (X-Y) = The intersection of X and Y

Proof Two: Using Venn Diagrams