Day 02- Discrete mathematics for computer science gate-Ir-reflexive Relation with basic understanding

Day 02- Discrete mathematics for computer science gate-Ir-reflexive Relation with basic understanding

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Day 02: Discrete Mathematics for Computer Science (GATE) – Irreflexive Relation with Basic Understanding

 What is a Relation in Discrete Mathematics?

In Discrete Mathematics, a relation is a set of ordered pairs that define a connection between elements of two sets. If a relation is defined on a single set, it is called a binary relation.

 What is an Irreflexive Relation?

A relation R on a set A is called irreflexive if no element in the set is related to itself. That is, for every element $a$ in $A$:

$$(a,a)\notin R$$

This means that self-loops are not allowed in an irreflexive relation.

 Example:

Let A = {1, 2, 3} and let R be a relation on A:

$$R=\{(1,2),(2,3),(3,1)\}$$

Since (1,1), (2,2), and (3,3) are NOT present in R, the relation is irreflexive.

 Difference Between Reflexive and Irreflexive Relations

Property	Reflexive Relation	Irreflexive Relation
Definition	Every element relates to itself	No element relates to itself
Condition	$(a,a)\in R$ for all $a\in A$	$(a,a)\notin R$ for all $a\in A$
Example	$R=\{(1,1),(2,2),(3,3),(1,2)\}$	$R=\{(1,2),(2,3),(3,1)\}$

 Special Cases:

A relation can be neither reflexive nor irreflexive if some but not all elements relate to themselves.
An empty relation is always irreflexive because it contains no pairs at all.

 Importance of Irreflexive Relations in Computer Science

 Used in graph theory for modeling directed graphs without self-loops.
 Important in order relations like strict partial order and strict preference relations.
 Helps in programming languages & automata theory for defining transitions without self-loops.

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Day 02- Discrete mathematics for computer science gate-Ir-reflexive Relation with basic understanding

Notes on Discrete Mathematics

Here is a clear and beginner-friendly explanation of Day 02 – Discrete Mathematics for Computer Science (GATE), focused on Irreflexive Relation, with basic understanding.

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Discrete Mathematics – Day 02 Topic: Irreflexive Relations

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What is a Relation? (Quick Recap)

In set theory, a relation is a subset of a Cartesian product of a set with itself.

For a set A, a relation R is a subset of A×AA \times A, meaning:

R⊆A×AR \subseteq A \times A

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Reflexive vs Irreflexive Relation

Reflexive Relation:

A relation RR on set AA is reflexive if:

∀a∈A, (a,a)∈R\forall a \in A,\ (a, a) \in R

That means every element is related to itself.

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Irreflexive Relation:

A relation RR on set AA is irreflexive if:

∀a∈A, (a,a)∉R\forall a \in A,\ (a, a) \notin R

That means no element is related to itself.

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Summary:

Property	Condition	Example
Reflexive	Every element maps to itself	(a, a) ∈ R for all a ∈ A
Irreflexive	No element maps to itself	(a, a) ∉ R for all a ∈ A

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Example of Irreflexive Relation

Let A={1,2,3}A = \{1, 2, 3\}

Let R={(1,2),(2,3),(3,1)}R = \{(1, 2), (2, 3), (3, 1)\}

Now, check:

• (1,1) ∉ R
• (2,2) ∉ R
• (3,3) ∉ R

So this is an irreflexive relation.

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Not Irreflexive Example

Let R={(1,1),(1,2),(2,3)}R = \{(1,1), (1,2), (2,3)\}

Here, (1,1) ∈ R → so it’s not irreflexive because at least one reflexive pair exists.

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Key Points to Remember

• A relation can be neither reflexive nor irreflexive.
• A relation cannot be both reflexive and irreflexive at the same time.
• Reflexive → All diagonal elements included
• Irreflexive → No diagonal elements allowed

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GATE/CS Exam Tip:

For multiple-choice questions:

• Always check for all (a,a)(a,a) for reflexivity
• If even one (a,a)∈R(a,a) \in R, then not irreflexive
• Use matrix or digraph methods if needed

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Summary Table:

Property	Meaning	Condition
Reflexive	All elements relate to themselves	∀ a ∈ A: (a, a) ∈ R
Irreflexive	No element relates to itself	∀ a ∈ A: (a, a) ∉ R

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Day 02- Discrete mathematics for computer science gate-Ir-reflexive Relation with basic understanding

Discrete Mathematics Using a Computer