Relations and Functions

In Class 11 you saw relation as set of ordered pairs. In Class 12, we study relation on same set. Example: On set of friends {A,B,C}, relation 'is brother of' may contain (A,B) if A is brother of B. Relation R is subset of A×A. Domain is set of first elements, range second. Example A={1,2,3}, R={(1,1),(2,2),(1,2)} is a relation on A. It tells who is connected to whom.

Reflexive means every element is related to itself. For all a∈A, (a,a)∈R. Think of 'is equal to' - every number equals itself. Example A={1,2,3}, R={(1,1),(2,2),(3,3),(1,2)} is reflexive because it has (1,1),(2,2),(3,3). If missing (2,2), not reflexive. Check reflexive: look for all (a,a).

Symmetric: If a related to b then b related to a. If (a,b)∈R then (b,a)∈R. Example 'is sibling of' - if A sibling of B then B sibling of A. If R has (1,2) but not (2,1), not symmetric. Transitive: If a related to b and b related to c then a related to c. If (a,b)∈R and (b,c)∈R then (a,c)∈R. Example 'is ancestor of' transitive. Example A={1,2,3} R={(1,2),(2,3),(1,3)} transitive.

Equivalence relation is that is reflexive, symmetric and transitive all three together. It divides set into groups (equivalence classes) where all elements in a group are equivalent. Example: On integers, relation 'has same remainder when divided by 3' - reflexive (a same remainder as itself), symmetric (if a same as b then b same as a), transitive (if a same as b and b same as c then a same as c). So equivalence. It partitions into 3 classes: remainder 0,1,2.

Function is special relation where each input has exactly one output. One-one (injective) means different inputs give different outputs, no two inputs share same output. Example f: N→N f(x)=2x is one-one because if 2x1=2x2 then x1=x2. But f: N→N f(x)=x² is not one-one on integers? On N it is one-one, on Z not because (-1)²=1². Test: f(a)=f(b) ⇒ a=b for one-one. Horizontal line test: graph cut at most once.

Onto (surjective) means every element in codomain B is hit by at least one element from domain. Range = Codomain. Example f: R→R f(x)=x³ is onto because every real y has x=cuberoot(y). But f: R→R f(x)=x² not onto because negative numbers in codomain never reached. Bijective = one-one and onto both together. Only bijective functions are invertible.

If f: A→B and g: B→C then gof: A→C defined as (gof)(x)=g(f(x)). First apply f then g. Order matters: fog ≠ gof generally. Example f(x)=2x, g(x)=x+3, fog(x)=f(g(x))=f(x+3)=2(x+3)=2x+6, gof(x)=g(f(x))=g(2x)=2x+3 different. Composition of two one-one is one-one, onto is onto, bijective is bijective.

Function f: A→B invertible if there exists g: B→A such that gof=IA and fog=IB. g is inverse f⁻¹. Condition: f is bijective iff invertible. Example f: R→R f(x)=2x+3 bijective, inverse find: y=2x+3 ⇒ x=(y-3)/2 so f⁻¹(y)=(y-3)/2. Check f⁻¹(f(x))=x. Graph of inverse is mirror over y=x. Only bijective functions have inverse.

Relations: 'is friend of' symmetric, 'is ancestor of' transitive, 'has same birthday' equivalence partitions people by birthday. Functions: Every student gets one roll number - function from students to roll numbers one-one. Aadhaar: person → Aadhaar number bijective. Netflix recommendation: user → movies. Composition: First convert Celsius to Fahrenheit then Fahrenheit to Kelvin is composition. Invertible: Encoding-decoding - encode message via function, decode via inverse. Biometric lock uses bijective mapping.