Sets and Functions in the Language of Proof
Set operations, images and preimages, and what injective, surjective and bijective actually require. Proofs at this level are mostly element-chasing arguments in this language.
What a learner can do afterwards
- Prove a set identity by double inclusion
- Prove a given function is injective or produce a counterexample
- Distinguish the image of a set from the preimage of a set
1 · Read
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.
Take f(x) equal to x squared on the reals. Then f(2) equals 4 and f(minus 2) equals 4, so two different inputs share one output and f is not injective. Now take g(x) equal to 2x plus 3. If g(a) equals g(b) then 2a plus 3 equals 2b plus 3, so 2a equals 2b and a equals b. Equal outputs force equal inputs, so g is injective. A function is surjective when every possible output value is hit, and bijective when it is both injective and surjective.
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.
Chase an arbitrary element both ways for equality, test injectivity input by input, and push forward for images while you pull back for preimages.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.