Chapter 2 - Relations and Functions

Relation links two sets. Before that ordered pair (a,b) where order matters. (1,2) is different from (2,1) because first element 1 second 2. Example: (Alex, Math) means Alex studies Math. (Math, Alex) is different nonsense Math studies Alex no meaning order matters.

Real life: Seat arrangement (Roll No, Seat No) ordered pair, (Student, Subject) enrollment, (Person, Phone Number) contact list.

Cartesian Product A×B = set of all possible ordered pairs (a,b) where a∈A b∈B. Example A={1,2} B={x,y} then A×B={(1,x),(1,y),(2,x),(2,y)} 2×2=4 pairs. Formula n(A×B)=n(A)×n(B). If A has 3 students and B has 3 subjects, A×B has 3×3=9 possible student-subject pairs table.

Relation R is subset of A×B R⊆A×B meaning we choose some pairs from all possible pairs that satisfy a condition. Example R = "is enrolled in" we choose only actual enrollments not all possible. Domain = set of all first elements used, Range = set of all second elements used.

A×B = {(a,b): a∈A b∈B} all ordered pairs. Properties: A×B ≠ B×A generally order matters unless A=B or one empty. Example A={1,2} B={3,4} A×B={(1,3),(1,4),(2,3),(2,4)} B×A={(3,1),(3,2),(4,1),(4,2)} different. n(A×B)=n(A)×n(B). If A has m elements B has n elements A×B has m×n elements.

A×B×C three sets ordered triple (a,b,c). Real life: A=shirts {red,blue} B=pants {jeans,shorts} A×B 4 outfits. A×A = A² = {(a1,a1),(a1,a2),(a2,a1),(a2,a2)}.

Important: Number of relations from A to B = 2^(n(A)×n(B)) because each pair can be either in relation or not two choices. Example A has 2 B has 3 A×B has 6 pairs relations =2^6=64 possible relations.

Relation R from A to B is subset of A×B R⊆A×B. Example A={1,2,3} B={a,b} A×B 6 pairs R={(1,a),(2,b)} relation choosing 2 pairs. Domain of R = set of all first elements {1,2} Range = set of all second elements {a,b} Codomain = B whole set {a,b} where relation goes.

Types: Empty relation φ no pair no connection, Universal relation A×B itself all pairs all connected. Example empty: no student enrolled no pair, universal: every student enrolled in every subject all 9 pairs.

Representation: roster list pairs {(1,2),(2,4)}, set-builder R={ (x,y): y=2x x∈A}, arrow diagram arrows from A to B, table.

Real life: Relation "is father of" set of ordered pairs (Father, Child), "is taller than" (Person1 taller than Person2).

Function is special relation where each element of domain has exactly one image in codomain unique output no element has two outputs. Definition: f: A→B is function if every a∈A has unique b∈B with (a,b)∈f. Example A={1,2,3} B={a,b} f={(1,a),(2,a),(3,b)} function each input one output. g={(1,a),(1,b),(2,a)} not function because 1 has two outputs a and b confusion not allowed.

Domain = set of all inputs where function defined A, Codomain = set B where output goes possible values, Range = set of actual outputs that occur subset of codomain Range⊆Codomain. Example f={(1,a),(2,a),(3,b)} Domain {1,2,3} Codomain {a,b} Range {a,b} actual outputs both a and b used. If f={(1,a),(2,a)} Domain {1,2} Codomain {a,b} Range {a} only a used b not used Range subset Codomain.

Vertical line test for graph: draw vertical line if it cuts graph at most once then function one output per input if cuts more than once not function.

Real life: vending machine function input money button → one snack unique output, each student has one roll number unique function Student→Roll No, each person has one Aadhaar number function Person→Aadhaar.

Equal functions: two functions equal if same domain and same output for each input f(x)=g(x) for all x.

One-one injective: different inputs give different outputs no two inputs have same output. Example f(x)=2x one-one because 1→2 2→4 different inputs different outputs. f(x)=x² not one-one because 1→1 -1→1 two inputs same output 1 and -1 both 1 many-one.

Many-one: two or more inputs same output. Example f(x)=x² many-one as above 1 and -1 same 1.

Onto surjective: every element of codomain has pre-image Range=Codomain all codomain used no element left unused. Example A={1,2} B={a,b} f={(1,a),(2,b)} onto Range {a,b}=Codomain {a,b} all used. If f={(1,a),(2,a)} Range {a}≠Codomain {a,b} b unused not onto into.

Into: there exists at least one element in codomain with no pre-image Range⊂Codomain proper subset some codomain unused not onto.

Bijective: both one-one and onto one-one and every codomain used perfect pairing.

Real life: one-one Student→Roll No different students different roll no, many-one Students→Birth Month many students same month, onto Tickets→Seats every seat has ticket if onto.

Real functions f: R→R real numbers to real numbers. Important graphs:

Identity function f(x)=x input same as output graph straight line 45° passing through origin slope 1 Domain R Range R.

Constant function f(x)=c fixed constant example f(x)=5 always 5 graph horizontal line y=5 Domain R Range {c} single value {5}.

Polynomial f(x)=a0+a1x+a2x²+... example f(x)=x² parabola U shape Domain R Range [0,∞) ≥0 because x²≥0.

Modulus absolute value f(x)=|x| = x if x≥0 -x if x<0 distance from 0 always ≥0 V shape graph Domain R Range [0,∞). Example |3|=3 |-3|=3 distance 3.

Signum function f(x)=1 if x>0 0 if x=0 -1 if x<0 sign of number positive 1 zero 0 negative -1 graph three horizontal lines Domain R Range {-1,0,1} only three values.

Greatest integer [x] or floor function greatest integer ≤x example [2.3]=2 [2.9]=2 [3]=3 [-1.2]=-2 greatest ≤-1.2 is -2 not -1 because -2≤-1.2<-1. Graph steps staircase. Domain R Range integers Z ...-2,-1,0,1,2...

Rational f(x)=1/x Domain R-{0} exclude 0 division by zero not allowed Range R-{0} hyperbola.

If f and g are functions with same domain D then we can add subtract multiply divide:

(f+g)(x)=f(x)+g(x) addition combine outputs Example f(x)=2x+3 g(x)=x-1 (f+g)(x)=2x+3+x-1=3x+2 Domain common Df∩Dg.

(f-g)(x)=f(x)-g(x) subtraction Example (2x+3)-(x-1)=2x+3-x+1=x+4.

(f·g)(x)=f(x)·g(x) multiplication Example (2x+3)(x-1)=2x·x+2x·(-1)+3·x+3·(-1)=2x²-2x+3x-3=2x²+x-3.

(f/g)(x)=f(x)/g(x) division with condition g(x)≠0 no division by zero Example (2x+3)/(x-1) with x≠1 because g(x)=x-1≠0 => x≠1 Domain Df∩Dg minus points where g(x)=0.

Composition fog and gof: (fog)(x)=f(g(x)) read f of g of x plug g(x) inside f. Example f(x)=2x+3 g(x)=x-1 fog = f(g(x)) = f(x-1)=2(x-1)+3=2x-2+3=2x+1. gof = g(f(x)) = g(2x+3)=(2x+3)-1=2x+2. Note fog≠gof generally composition not commutative order matters.

Domain of fog: {x: x∈Dg and g(x)∈Df} x must be in domain of g and g(x) must be in domain of f.

Short tricks:

n(A×B)=n(A)×n(B) and n(A×B×C)=n(A)×n(B)×n(C).

Number of relations from A to B =2^(n(A)×n(B)) each pair two choices in or not.

Number of functions from A to B = n(B)^n(A) each element of A has n(B) choices for image multiply n(A) times. Example A 2 elements B 3 elements functions 3²=9.

Domain Range: For f(x)=√(x-a) Domain x≥a because inside root ≥0, Range ≥0. For f(x)=1/(x-a) Domain x≠a because denominator ≠0.

Modulus trick: |x|≥0 always, |x|=|-x|, |x|²=x², |x+y|≤|x|+|y| triangle inequality.

Greatest integer: [x]≤x<[x]+1, [x]+[-x]= -1 if x not integer 0 if x integer, [x+n]=[x]+n if n integer.

Real life: Function machine input→output vending machine money→snack, ATM pin→money, student→percentage, temperature→weather, distance→fare taxi fare function of distance, time→growth plant height function of time.