function and relation

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!