This document defines and provides examples of different types of relations and functions. It begins by defining what a relation is as a set of ordered pairs where one element is related to another. It then defines various types of relations including reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and transitive relations. Next, it defines what a function is as a special type of relation where each input is mapped to exactly one output. It discusses the domain, codomain, range, image, and preimage of a function. Finally, it defines injective, surjective, bijective functions as well as inverse functions and function composition.

1. Function and Relation
Prepared by: Angela Clarito

2. Relation
● It is a Rule that pairs each element in one set.
● It is a set of ordered pairs, x and y.
● A relation R on a set X is a subset of X x X. If (a,b) ∊ R,
we write x R y. Reads as “ x is related to y”.

3. Relation
● A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3))
X y
2
1 1
2
3
R

4. Types of Relations

5. Reflexive
Irreflexive
Symmetric
Asymmetric
Antisymmetric
Transitive

6. Reflexive Relation
● A relation R on a set is called reflexive if (a,a) ∊ R for every
element a ∊ A. In other words, ∀ a((a,a) ∊ R).
● A relation will be known as reflexive relative if every element of
set A is related to itself. The word reflexive means that in a set,
the image of every element has its own reflection.
Example:
Let A = {1,2,3,4}
R={(1,1),(2,2),(3,3),(4,4)}

7. Irreflexive Relation
● A relation R on a set A is called irreflexive if ∀a ∊ A,
(a,a) ∉ R.
Example:
Let A = {1,2,3,4}
R={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),
(3,4),(4,1),(4,2),(4,3)}

8. Symmetric Relation
● A relation R on a set A is called symmetric if (b,a) ∊ R
holds when (a,b) ∊ R for all a, b ∊ A.
● In other words, relation R on a set A is symmetric if
∀a ∀b((a,b) ∊ R → (b,a) ∊ R.
Example:
Let A = {1,2,3,4}
R={(1,2),(2,1),(1,3),(3,1),(1,4),(4,1)}

9. Antisymmetric Relation
● A relation R on a set A is called antisymmetric if ∀a
∀b((a,b) ∊ R ^ (b,a) ∊ R → (a=b)) whenever we have
(a,b) in R, we will never have (b,a) in R until or unless
(a=b).
Example:
Let A = {1,2}
R1
={(1,1),(2,1),(2,2)}
R2
={(1,1),(1,2),(2,1),(2,2)}

10. Asymmetric Relation
● A relation R on a set A is called asymmetric if ∀a
∀b((a,b) ∊ R ^ (b,a) ∉ R) whenever we have (a,b) in R,
we will never have (b,a) in R until or unless (a=b).
Example:
Let A = {1,2,3,4}
R={(1,2),(1,3),(1,4),(2,3),(2,4)}

11. Transitive Relation
● A relation R on a set A is called transitive if ∀a ∀b
∀c(((a,b) ∊ R ^ (b,c) ∊ R) → (a,c) ∊ R)
Example:
Let A = {1,2,3}
R={(1,3),(1,2),(3,2)}

12. Function
● It is a Rule that pairs each x-coordinate to exactly one element
from y-coordinates.
● It is a set of ordered pairs (relation) where x-coordinate should
not be repeated.
● Are sometimes called as mappings or transformations.
● A function f from A to B , denoted f: A → B, assign each
element of A exactly on element of B.

13. Relation
● A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3))
X y
2
1 1
2
3
R

14. Terminologies

15. Domain
Codomain
Image
Preimage
Range

16. Domain
Is the set x.
Codomain
Is the set y .
Range
The values in set y that is paired. .

17. Preimage
The value in y that is paired to the
value in x.
Image
The value in x that is paired to the
value in y.

18. Answer the following
for f: X → Y
Domain
Codomain
Range
Image(s)
Preimage
f(y) = ?

19. Types of Functions

20. Injective
Surjective
Bijective
Inverse
Composition

21. Injective (One to one) Functions
Each Value in the range corresponds to exactly one element in
then domain.
∀ a ∀ b (( a ≠ b) → f (a) ≠ f (b)))
∀ a ∀ b ((f (a) = f (b)) → (a = b))
∀ reads as “for all”
≠ reads as “not equal to”
CONTRAPOSITIVE

22. Injective (One to one) Functions

23. Surjective (Onto) Functions
Each element in the codomain maps to at least one element in the
domain.
∀ y ∃ x (f(x) = y)
∃ reads as “ there exists”

24. Surjective (Onto) Functions

25. Bijective (One to one correspondence)
Functions
Functions that are both one-to-one and onto, or both surjective
and injective.
∀ y ∃ x (f(x) = y)
∃ reads as “ there exists”

26. Bijective (One to one correspondence)
Functions

27. Problem:
Let f be a function from X = { a, b, c, d } to Y = { 1, 2, 3 } defined by
f(a) = 3, f(b) = 2, f(c) = 2 and f(d) = 3. Is f: X → Y either one-to one
or onto?

28. Examples of Different Types of
Correspondences

29. One-to-one, not onto Onto, not one-to-one

30. One-to-one, onto Neither one-to-one, onto

31. Not a Function

32. Inverse Function
● Given any function, f, the inverse of the function f-1,
is a relation
that is formed by interchanging each (x, y) of f to a (y, x) of f-1,
.
● The function f would be denoted as f-1,
and read as “f inverse”.
NOTE:
A one-to-one correspondence is called invertible because we can define
an inverse of this function. A function is not invertible if it is not a one-to
one correspondence, because inverse of such a function does not exist.

33. Inverse Function

34. Composition of Function
● The composition of two functions g: A → B and f: B → C, denoted
by f о g, is defined by (f о g) (a) = f(g(a)) or (g о f) (a) = g(f(a))
This means that
First, function g is applied to element A, mapping it into an element of B.
Then, function f is applied to this element of B, mapping it into an element
of C.
Therefore, the composite function maps from A to C.

35. Composition of Function

36. Thank You!