This document discusses relations and functions for class 12. It defines concepts like cartesian product, relations, domain and codomain, range, types of relations such as reflexive, symmetric, transitive and equivalence relations. It also defines functions, one-to-one and onto functions, bijective functions, composition of functions, and inverse functions. An example is provided to show that a given relation is reflexive but not symmetric or transitive. Another example shows a given function is one-to-one but not onto.

1. RELATIONS AND FUNCTIONS
FOR CLASS XII
BY C SANJEIVE KUMAR

2. CARTESIAN PRODUCT
Let A and B be two non-empty sets . The cartesian
product A × B is the
set of all ordered pairs of elements from A and B

3. RELATION
A relation R from a non-empty set A to a non-empty set B
is a subset of the cartesian product A × B

4. DOMAIN AND CODOMAIN
The set of all first elements of the
ordered pairs in a relation R from a set
A to a set B is called the domain of the
relation R.
The whole set B is called the codomain
of the relation R.

5. RANGE
The set of all second elements in a relation R from a set A
to a set B is called the range of the relation R.
Range ⊆ codomain

6. Example:
Let A = {1, 2, 3, 4, 5, 6}. Define a
relation R from A to A by
R = {(x, y) : y = x + 1 }
(i) Write this relation as Roster form
(ii) Write down the domain,
codomain and range of R.

7. Answer:
Roster form :R = {(1,2), (2,3), (3,4), (4,5),
(5,6)}.
domain ={1, 2, 3, 4, 5,}
range = {2, 3, 4, 5, 6}
codomain = {1, 2, 3, 4, 5, 6}.

8. TYPES OF RELATIONS
A relation R in a set A is called empty relation, if no
element of A is related to any element of A

9. A relation R in a set A is called
universal relation, if each element
of A
is related to every element of A,
i.e., R = A × A

10. A relation R in a set A is called
Reflexive
if (a, a) ∈ R, for every a ∈ A,

11. Symmetric
if (a , b ) ∈ R implies that (b , a ) ∈ R, for all
a , b ∈ A

12. Transitive
if (a , b ) ∈ R and (b , c )
∈ R implies that (a , c ) ∈
R, for all a , b ,c ∈ R

13. A relation R in a set A is said to be
an equivalence relation if R is
reflexive, symmetric and transitive

14. Example:
Show that the relation R in the set
{1, 2, 3} given by R = {(1, 1), (2, 2), (3,
3), (1, 2), (2, 3)} is reflexive but
neither symmetric nor transitive.

15. Answer:
R is reflexive, since (1, 1), (2, 2)
and (3, 3) lie in R.
R is not symmetric, as (1, 2) ∈ R
but (2, 1) ∉ R.
R is not transitive, as (1, 2) ∈ R
and (2, 3) ∈ R
but (1, 3) ∉ R.

16. FUNCTION
A relation f from a set A to a set B is said to be a function
if every element of set A has one and only one image in
set B

17. If f is a function from A to B and (a,
b) ∈ f, then f (a) = b, where b is
called the
image of a under f and a is called
the pre image of b under f.

18. The function f from A to B is
denoted by f: A→ B.
A function which has either R (set of real numbers)or
one of its subsets as its range is called a real valued
function.
If its domain is also either R or a subset of R, it is called a
real function

19. ONE-ONE OR INJECTIVE
A function f : X → Y is defined to be one-one (or injective),
if the images
of distinct elements of X under f are distinct.

20. Onto or surjective
A function f : X → Y is said to be onto (or surjective), if
every element of Y is the image of some element of X
under f,

21. Bijective
A function f : X → Y is said to be one-one and onto (or
bijective), if f is both one-one and onto

22. Example:
Show that the function f : N → N, given by f
(x) = 2x, is one-one but not onto.
The function f is one-one, for f (x ) = f (y )
⇒ 2x=2y ⇒ x=y
f is not onto, as for 1 ∈ N( codomain), there does not
exist any x in N(domain) such that
f (x) = 2x = 1

23. Composition of Functions and
Invertible Function
Let f : A → B and g : B → C be two functions. Then the
composition of
f and g, denoted by gof, is defined as the function
gof : A → C given by
gof (x) = g(f (x)), ∀ x ∈ A.

24. Example: Let f : {2, 3, 4, 5} → {3, 4,
5, 9} and g : {3, 4, 5, 9} → {7, 11, 15}
be functions defined as f (2) = 3, f
(3) = 4, f (4) = f (5) = 5 and g (3) = g
(4) = 7 and g (5) = g (9) = 11. Find
gof.

25. Answer:
gof (2) = g (f (2)) = g (3) =7
gof (3) = g (f (3)) = g (4) = 7,
gof (4) = g (f (4)) = g (5) = 11
and gof (5) = g (5) = 11.

26. Inverse
A function f : X → Y is defined to be invertible, if there
exists a function g : Y → X such that gof = I𝑿 and fog = IY.
The function g is called the inverse of f and is denoted by
𝒇−𝟏.

27. Example: Let f : N → Y be a
function defined as f (x) = 4x + 3,
where,
Y = {y ∈ N :y = 4x + 3 for some x ∈
N}. Show that f is invertible. Find
the inverse.

28. THANKS