DISCRETE MATHS (Relations and Functions), Lecture notes of Discrete Mathematics

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RELATIONS &

FUNCTIONS

DIPLOMA IN INFORMATION TECHNOLOGY

DIT 1113

DISCRETE MATEMATICS

Expected Outcome

• (^) Understand what is Relation

Understand what is are the properties of relations

• (^) Understand what is a Funtion

Properties of Relations

• (^) Reflexive
• (^) Irreflexive
• (^) Symmetric
• (^) Anti-Symmetric
• Transitive

Reflexive/ Irreflexive Properties

• A relation R on a set A is said to be Reflexive

If ,

R is said to be reflexive, if ( , ) ∈ R for all ∈ A that is,

every element of A is R-related to itself, in other words

R for every ∈ A.

Reflexive: The relation R on {1,2,3} given by R =

{(1,1), (2,2), (2,3), (3,3)} is reflexive. (All loops are

present.)

• A relation R on a set is said to be irreflexive

If ,

Symmetric Relation

• (^) A relation R on a set A is said to be Symmetric

If ,

(a, b)

(b, a)

R is symmetric if for all , ∈ A, ( , ) ∈ R implies, (, )

∈ R. (R implies that R)

The relation R on {1,2,3} given by R = {(1,1), (1,2),

(2,1), (1,3), (3,1)} is symmetric. (All paths are 2-way

Antisymmetry/ Asymmetric

Relation

• (^) Anti-symmetry and asymmetry is against symmetry.
• (^) Anti-symmetry – A relation R on a set A is said to be

anti -symmetric

,

if (a, b)

(b, a)

a = b

• (^) Asymmetry – A relation R on a set A is said to be

asymmetric

∀ 𝑎, 𝑏∈𝐴 , ,

if (a, b) ∈𝑅

(b, a) 𝑅

Violations of the Properties

• (^) Why is R = {(1,1), (2,2), (3,3)} not reflexive on {1,2,3,4}?

Because (4,4) is missing.

• Why is R = {(1,2), (2,1), (3,1)} not symmetric?

Because (1,3) is missing.

• Why is R = {(1,2), (2,3), (1,3), (2,1)} not transitive?

Because (1,1) and (2,2) are missing.

• Is {(1,1), (2,2), (3,3)} symmetric? transitive?

Yes!

Equivalence Relation

• (^) A relation R on a set A is said to be equivalent

relation if R is

Reflexive

Symmetric

Transitive

FUNCTIONS

Q: Is this mapping a function?

One to One Function

A function f is called one-to-one if and only if f (x) = f (y)

implies x = y for every x,y in the domain of f :

∀x,y. ( f (x) = f (y) → x = y)

• (^) One-to-one functions never assign different elements in

the domain to the same element in the codomain:

∀x,y. (x ≠ y → f (x) ≠ f (y))

• (^) A one-to-one function also called injection or injective

function

FUNCTION COMPOSITION