This document defines basic concepts related to relations and functions. It defines what a relation is between two sets and provides examples of different types of relations including reflexive, symmetric, transitive, and equivalence relations. It also defines one-to-one, onto, bijective, and invertible functions. Several short questions are provided as examples related to these concepts along with longer proof-based questions.

1. RELATIONS& FUNCTIONS
Basic concepts and formulae
 Relation : If A and B are two non- empty sets, then any subset R of A ×B is called relation from set A to set B.
i.e. R : A → 𝐵 ⇔ 𝑅 ⊆ A × B.
 Some standard types of Relations : Let A be a non- empty set. Then, a relation R on set A is said to be.
 Reflexive : If (x, x)∈ R for each element x∈ A, i.e., if x R x for each element x ∈ A.
 Symmetric : If (x, y)∈ R ⇒ (y, x) ∈ R for all x, y∈ A, i.e., if x Ry ⇒ y Rx for all x, y ∈ A.
 Transitive : If (x, y)∈ R and (y, z)∈ R ⇒ (x, z) ∈, for all x, y, z ∈ A, i.e., if xRy and yRz ⇒ xRz.
 Equivalence relation : Any relation R on a set A is said to be an equivalence relation if R is reflexive,
symmetric and transitive.
 A function F:X→Y is one -one or injective if f(x1)=f(x2) ⟹x1= x2 for all x1, x2∈ X
 A function F:X→Y is onto or surjective if for every y∈ Y there exist x∈ X: f(x)=y
 A function which is both one-one and onto is called bijective.
 A function F:X→Y is invertible if and only if f is bijective.
 The composition of functions f: A→B and g: B→C is the functiopn gof: A→C given by gof(x)= g(f(x))
 A function F:X→Y is invertible if there exist g: Y→X such thast gof=Ix and fog= Iy.
Short Questions
1. What is the range of the function f(x) =
𝑥−1
𝑥−1
2. F: R→R f(x)= (3-x3
)1/3
, find (fof)(x)
3. If f(x) = x2
+ 2 and g(x) =
x
x+1
, find gof (5).
4. F : R→R is defined by f(x)= 3x+2. Find f(f(x))
5. A= {1,2,3} and B={4,5,6,7} and f={(1,4),(2,5),(3,6)} be a function from A to B. state whether f is one one or onto.
6. Given an example to show the relation R in the set of real nos. , defined by R = {( x, y); x≤ y2
} is not transitive.
7. Let f : R - {
−3
5
} → R be a function defined as f (x) =
2x
5x+3
, find f-1
.
8. If f(x) = 2x + 5 , g(x) = 2x – 5 , x∈ R find (fog) (9).
9. If R = {(x , y) : x2
+ y2
≤ 4 ; x , y∈Z } is a relation in Z , write the domain of R.
10. If f : R→R be a function defined by f(x) = 3x – 4 , then write f-1
(x).
11. f : R – (- 1) → R – (+ 1) be defined as f(x) =
x
x+1
, find f -1
(x).
12. Find the smallest equivalence Relation on the set A={1,2,3}
13. Show that modulus function F : R→R is not one –one and onto.
Long Questions
1. Prove that f: N→N defined by f(x)= x2
+x+1 is one- one but not onto.
2. Prove that the relation R in set A={5,6,7,8,9} given by R= {(a,b): |a-b| is divisible by 2} is an equivalence relation.
Find all the elements related to element 6.
3. Let T be the set of all triangles in a plane with R as a relation in T given by R = {(T1 , T2) : T1≅ T2}. Show that R
is an equivalence relation.
4. Show that the relation R on Z defined by R = {(a, b) : a – b is divisible by 5} is an equivalence relation.
5. Let N be the set of all natural numbers and R be the relation in N ×N defined by
(a , b) R (c , d) if ad = bc. Show that R is an equivalence relation.
6. Show that the function f : R → R defined by f(x) = 2x3
– 7 for x ∈ R is bijective.

2. 7. Let A={1,2,3......9} be the set of all natural numbers and R be the relation in A ×A defined by
(a , b) R (c , d) if a+d = b+c. Show that R is an equivalence relation. Also obtain the equivalence class [(2,5)]
8. Let N be the set of all natural numbers and R be the relation in N ×N defined by
(a , b) R (c , d) if ad(b+c) = bc(a+d). Show that R is an equivalence relation.
9. Show that the relation R in the set : A = { x : x ∈ Z, 0≤x ≤12} given by.
R = {( a, b): a − b is divisible by 4} is an equivalence Relation.
10. Show that the relation R in the set R of real no. defined as R = {( a, b) : a ≤ b2
} is neither reflexive nor symmetric
nor transitive.
11. Prove that the relation R in the set A = { 1, 2, 3, 4, 5} given by R = {(a, b) : a − b is even } is an equivalence
relation. Also obtain the equivalence class of {1}
12. Let A = R – { 3 } and B = R – { 1}, f : A → B defined by f (x) =
x−2
x−3
. Is f is one – one and onto ? Justify your
answer.
13. Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
14. Let f : R →R be defined as f(x) = 10x + 7. Find the function g : R → R : gof = fog = IR.
15. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {( L1 , L2 ) : L1 is parallel to L2 }.
Show that R is an equivalence relation.
16. Check whether the relation R in R defines by R = {(a, b) : a ≤ b3
} is reflexive, symmetric or transitive.
17. Show that the relation R in the set Z of integers given by R = {( a, b) : 2 divides a – b} is an equivalence relation.
18. State whether the function is one – one, onto f : R → R : f (x) = 1 + x2
. Justify your answer.
19. Show that f : [ -1 , 1 ] →R, given by f(x) =
x
x+2
is one – one. Also find f-1
.
20. Let f : N → R be a function defined as f (x) = 4x 2
+ 12x + 5. Show that f : N → S is invertible and find the inverse
of f , where S is the range of the function
21. Consider f : R+→ [ 4, ∞ ) given by f (x) = x2
+ 4. Show that f is invertible with f -1
(y) = y − 4.
22. Consider f : R+→ [ -5 , ∞ ) given by f (x) = 9x2
+ 6x - 5. Show that f is invertible with f -1
(y) =
y+6−1
3
.
23. Show that if f : R -
7
5
→ R -
3
5
is defined by f(x) =
3x+4
5x−7
and g : R -
3
5
→R -
7
5
is defined by
g(x) =
7x+4
5x−3
, then fog = IA and gof = I B where IA (x) = x, IB (x) = x.
24. Show that the relation R on the set A = {x ∈ Z : 0≤ 𝑥 ≤ 12}, given by
R = {(a, b): 𝑎 − 𝑏 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 of 4} is an equivalence relation.
25. Let A = R – {3} and B = R -
2
3
. If f : A → B: f(x) =
2x−4
3x−9
, then prove that f is a bijective function.
26. Consider f : R+→ [ -9 , ∞ ) given by f (x) = 5x2
+ 6x - 9. Show that f is invertible with f -1
(y) =
54+5y−3
5
.
27. Let A = R – { b } and B = R – { 1}, f : A → B defined by f (x) =
x−a
x−b
. show that f is bijective function