Sets and Functions in the Language of Proof

Two sets are equal exactly when they hold the same elements. To prove an identity, chase an arbitrary element in both directions. First take any x on the left side and show it lies on the right side. Then take any x on the right side and show it lies on the left side. Union, intersection and complement combine sets the way or, and and not combine statements. Sketch the two circle Venn picture to guess the identity before you write the chase.

The image of a set collects the outputs the function produces from that set. With f(x) equal to x squared, the image of the set containing minus 2 and 3 is the set containing 4 and 9. The preimage of a set collects every input whose output lands inside it. The preimage of the set containing 4 is the set containing minus 2 and 2. A preimage can be larger than its target, since both 3 and minus 3 square to 9.

When you meet a new claim, keep one small example on paper and test each definition against it. A composition of two injective functions is injective. Suppose f and g are injective and f(g(a)) equals f(g(b)). The outer function gives g(a) equals g(b), and the inner function gives a equals b. Equal outputs force equal inputs through both layers, so the composition passes the test.

Run the injectivity test on paper. Suppose g(a) equals g(b), so 2a plus 3 equals 2b plus 3. Subtract 3 to get 2a equals 2b, then divide by 2 to reach a equals b.

Stopping halfway earns nothing. Lee showed left inside right and claimed equality, but the reverse containment is still unproved. Name the missing half before claiming done.