Relations and Functions
Board: CBSE |Class:Class11
ComprehensivestudynotesforRelationsandFunctionsbyAjayYadav(MathKingofKatargam).Mastereveryconceptwithclearexplanations,solvedexamples,andpracticeproblems.
A×B={(a,b):a&inA,b&inB}.OrderedpairswherefirstelementfromA,secondfromB.n(A×B)=n(A)×n(B).
ArelationRfromAtoBisasubsetofA×B.If(a,b)&inR,wewriteaRb.Domain:Setoffirstelements.Range:Setofsecondelements.
Reflexive:(a,a)&inRforalla.Symmetric:If(a,b)&inRthen(b,a)&inR.Transitive:If(a,b),(b,c)&inRthen(a,c)&inR.Equivalence:Reflexive+Symmetric+Transitive.
Afunctionf:A→BassignseachelementofAtoexactlyoneelementofB.Domain:A.Codomain:B.Range:{f(a):a&inA}.
One-to-one(Injective):f(a)=f(b)⇒a=b.Onto(Surjective):Range=Codomain.Bijective:Bothone-to-oneandonto.
Keyfunctions:Identityf(x)=x,Constantf(x)=c,Polynomial,Rational,Modulusf(x)=|x|, Signum, Greatest Integer f(x)=[x].
Important Formulas
| Cartesian Product | A × B = {(a,b): a∈A, b∈B} |
| Domain | Set of all first elements of ordered pairs |
| Range | Set of all second elements of ordered pairs |
| Reflexive | (a,a) ∈ R for all a ∈ A |
| One-to-one | f(a)=f(b) ⇒ a=b |
| Onto | Range = Codomain |
Solved Examples
Example 1: If A = {1,2}, B = {a,b}, find A × B.
Solution: A × B = {(1,a), (1,b), (2,a), (2,b)}.
Example 2: Check if f(x) = x² is one-to-one on R.
Solution: f(1) = f(-1) = 1, but 1 ≠ -1. Not one-to-one on R.
Example 3: Find domain and range of f(x) = √x-2.
Solution: Domain: x ≥ 2 or [2, ∞). Range: [0, ∞).
Practice Questions
- Find A × B × C if A={1,2}, B={a}, C={x,y}.
- Check if R = {(1,2), (2,3), (1,3)} on {1,2,3} is transitive.
- Find domain and range of f(x) = 1/(x-3).
- Show that f(x) = 2x+3 is bijective from R to R.
- If f(x) = x² + 1 and g(x) = 2x – 3, find fog and gof.
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