Relations and Functions: Products, Relations, and Composition
What the exam tests
This domain checks whether you can build sets of ordered pairs, classify relations, and decide when a relation is a function. You also need to compose functions and read domain and range from a rule.
Core rules
- The Cartesian product A×B = {(a,b): a∈A, b∈B}; n(A×B)=n(A)n(B). Order matters, so A×B≠B×A unless A=B.
- A relation from A to B is any subset of A×B. A function from A to B assigns exactly one b to each a; every function is a relation, but not every relation is a function.
- Types: reflexive (aRa), symmetric (aRb⇒bRa), transitive (aRb and bRc⇒aRc), equivalence (all three). A function is one-one if different inputs give different outputs, onto if every codomain element is hit.
- Composition: (f∘g)(x)=f(g(x)); apply g first. Domain of f∘g requires x in domain of g and g(x) in domain of f.
Worked example
Let f(x)=2x+1 and g(x)=x². Find f∘g and g∘f at x=3. (f∘g)(3)=f(g(3))=f(9)=19. (g∘f)(3)=g(f(3))=g(7)=49. The order changes the result, so composition is not generally commutative.
Traps
Writing (3,5) as (5,3); treating a many-to-one relation as a function; forgetting that a function’s range can be smaller than its codomain; and applying f before g in f∘g.