Here is a clear, complete, and exam-oriented explanation of Introduction to Relations in Discrete Structures.

⭐ Introduction to Relations

A relation is a fundamental concept in discrete mathematics, computer science, and database theory.
It describes how elements of one set are related to elements of the same or another set.

✔️ Formal Definition

Let A and B be two sets.

A relation R from A to B is defined as:

[
R \subseteq A \times B
]

That means:

A relation is a subset of the Cartesian product of A and B.
A relation consists of ordered pairs (a, b) where
( a \in A ) and ( b \in B ), and a is related to b.

Example:

Let
A = {1, 2, 3}
B = {x, y}

A relation R from A to B may be:

[
R = {(1,x), (2,y)}
]

Here,

1 is related to x
2 is related to y

⭐ Relation on a Set (A to A)

If A = B, then relation is called a relation on set A.

Example:
A = {1, 2, 3}

Possible relation R:

[
R = {(1,1), (2,3), (3,3)}
]

⭐ How Relation is Formed?

Steps:

Take two sets A and B
Form all ordered pairs (Cartesian product)
Choose any subset of A × B — this subset is a relation

⭐ Types of Relations (Overview)

These come after introduction, but listing them helps:

Reflexive Relation
Symmetric Relation
Transitive Relation
Equivalence Relation
Antisymmetric Relation
Partial Order Relation

(You can ask for detailed explanation of these.)

⭐ Representation of Relations

Relations can be represented in several ways:

1. Set of Ordered Pairs

Example:
R = {(1,2), (2,3), (3,3)}

2. Arrow Diagram

Draw arrows from elements of A to B.

3. Matrix Representation

If A = {a₁, a₂} and B = {b₁, b₂}, then relation can be shown as:

	b₁	b₂
a₁	1	0
a₂	0	1

1 means related; 0 means not related.

4. Directed Graph (Digraph)

For relations on a set A, we use nodes and directed edges.

⭐ Real-Life Examples of Relations

✔ Student–Course Relation

R = {(student, course)}

✔ Employee–Manager Relation

(employee, manager)

✔ Precedence Relation (≤)

(2 ≤ 5)

✔ Database table rows

Each row is a set of related values → a relation.

⭐ Domain and Range of a Relation

For a relation
[
R \subseteq A \times B
]

Domain(R): all first elements of ordered pairs
Range(R): all second elements of ordered pairs

Example:

R = {(1,2), (3,4), (5,2)}

Domain = {1, 3, 5}
Range = {2, 4}

⭐ Important Observations

A relation can contain any number of ordered pairs
It can be empty
Maximum number of relations from A to B is:
[
2^{|A| \times |B|}
]