Part 01-Discrete mathematics for gate-Partial Order Relations and it’s matrix representation

Part 01-Discrete mathematics for gate-Partial Order Relations and it’s matrix representation

Partial Order Relations and Its Matrix Representation (Discrete Mathematics for GATE)

1. Introduction to Partial Order Relations
A partial order relation (poset) is a binary relation $R$ on a set $S$ that satisfies the following properties:

• Reflexivity: $aRa$ for all $a\in S$

• Antisymmetry: If $aRb$ and $bRa$, then $a=b$

• Transitivity: If $aRb$ and $bRc$, then $aRc$

A set $S$ along with a partial order relation $R$ is called a partially ordered set (poset).

2. Matrix Representation of Partial Order Relations
The relation matrix of a partial order relation on a finite set is a square matrix MRM_RMR​ of size $n\times n$, where:

• Mij=1M_{ij} = 1Mij​=1 if aiRaja_i R a_jai​Raj​

• Mij=0M_{ij} = 0Mij​=0 otherwise

Example:
Let $S=\{1,2,3\}$ with a relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}$.

The relation matrix MRM_RMR​ is:

MR=[110011001]M_R = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}MR​=​100​110​011​​

• The diagonal elements are 1, ensuring reflexivity.

• The matrix is not symmetric, proving antisymmetry.

• The transitive closure can be found using the Warshall’s Algorithm.

3. Hasse Diagram Representation
A Hasse diagram is a graphical representation of a poset where:

• Reflexive edges are omitted.

• Transitive edges are removed for simplicity.

• Elements are arranged hierarchically.

For the above example, the Hasse Diagram is:

4. Applications of Partial Order Relations

• Scheduling Problems

• Hierarchical Structures

• Dependency Graphs in Computer Science

• Data Organization in Databases

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Part 01-Discrete mathematics for gate-Partial Order Relations and it’s matrix representation

DIGITAL NOTES ON Discrete Mathematics B.TECH II YEAR

Discrete Mathematical Structures Unit-1

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Part 01 – Discrete Mathematics for GATE

Topic: Partial Order Relations and Its Matrix Representation

Language: English (can also be provided in Hindi upon request)
Useful for: GATE, B.Tech, UGC-NET, BCA, MCA, Competitive CS Exams

1. What is a Relation in Set Theory?

A relation $R$ from a set $A$ to $A$ is a subset of $A\times A$, i.e., a set of ordered pairs.

Example:
Let $A=\{1,2,3\}$
A relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}$

2. What is a Partial Order Relation?

A Partial Order Relation on a set $A$ is a binary relation $R\subseteq A\times A$ that is:

1. Reflexive: $(a,a)\in R$ ∀ $a\in A$

2. Antisymmetric: $(a,b)\in R$ and $(b,a)\in R$ ⟹ $a=b$

3. Transitive: $(a,b)\in R$ and $(b,c)\in R$ ⟹ $(a,c)\in R$

If a relation satisfies these three properties, it is called a partial order, and the set $A$ with relation $R$ is called a partially ordered set (poset).

3. Matrix Representation of a Relation

To represent a relation $R$ on set A={a1,a2,…,an}A = \{a_1, a_2, …, a_n\}A={a1​,a2​,…,an​} as a matrix:

• Create an $n\times n$ matrix MRM_RMR​

• Mij=1M_{ij} = 1Mij​=1 if (ai,aj)∈R(a_i, a_j) \in R(ai​,aj​)∈R, else Mij=0M_{ij} = 0Mij​=0

Example:

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

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

Matrix Representation:

4. Checking Properties via Matrix

5. GATE Perspective Questions:

1. Identify whether a relation is partial order

2. Find the matrix of a given relation

3. Check whether a relation is reflexive, antisymmetric, transitive

4. Count number of relations with given properties

Summary

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Part 01-Discrete mathematics for gate-Partial Order Relations and it’s matrix representation

Bernard Kolman_ Robert C. Busby_ Sharon Cutler Ross- …