Relations and Functions - BSEB Inter Study Guide
Relations and Functions
BSEB Inter Mathematics tests whether you can classify relations, prove a function is one-one or onto, compose functions and invert bijections. Expect short proofs and domain checks rather than long computation.
Core rules
- A relation R on A is reflexive if (a,a)∈R for all a; symmetric if (a,b)∈R implies (b,a)∈R; transitive if (a,b),(b,c)∈R implies (a,c)∈R. All three make an equivalence relation.
- f:A→B is one-one if f(x1)=f(x2) implies x1=x2; onto if every y∈B has some x∈A with f(x)=y; bijective if both.
- Only a bijection has an inverse, and (f⁻¹∘f)(x)=x, (f∘f⁻¹)(y)=y.
- Composition is not commutative in general: fog is not equal to gof.
Worked example
Show f:R→R, f(x)=2x+3 is bijective and find f⁻¹. One-one: 2x1+3=2x2+3 implies x1=x2. Onto: for y∈R take x=(y−3)/2; then f(x)=y. So f is bijective. Swap x and y: x=2y+3 gives y=(x−3)/2. Thus f⁻¹(x)=(x−3)/2.
Traps that cost marks
- Checking one-one on the wrong domain: x² is one-one on [0,∞) but not on R.
- Confusing codomain with range; onto is about the stated codomain.
- Forgetting to verify all three properties for an equivalence relation.
- Writing f⁻¹(x)=1/f(x), which is false.