Relations and Functions
Relations and Functions tests whether you can classify relations, prove equivalence, and decide when a function is one-one, onto, or invertible. The exam also uses composition and binary operations, so definitions must be stated precisely.
Core rules
A relation from A to B is any subset of A×B. Its domain is the set of first entries, range the set of second entries, and codomain is B. An equivalence relation is reflexive, symmetric and transitive. A function f:A→B is one-one if f(x₁)=f(x₂) implies x₁=x₂; it is onto if every element of B is an image. Composition (f∘g)(x)=f(g(x)) is defined only when the codomain of g meets the domain of f. An inverse exists exactly when f is bijective, and then f⁻¹(y)=x ⇔ f(x)=y.
Worked example
Show that f:R→R, f(x)=2x+3 is bijective and find its inverse. If f(a)=f(b), then 2a+3=2b+3, so a=b; hence one-one. For any y∈R, choose x=(y−3)/2. Then f(x)=2((y−3)/2)+3=y, so onto. Therefore f⁻¹(y)=(y−3)/2.
Traps
Do not confuse range with codomain. A relation can be symmetric without being reflexive. Composition is not generally commutative: f∘g≠g∘f. An inverse may be written only after proving bijectivity; writing a formula is not enough. For binary operations, check closure before checking associativity or identity.