MATRIX REPRESENTATION OF A RELATION.pptx
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The document discusses relation matrices, which are zero-one matrices that can represent relations between finite sets. A relation matrix has rows and columns corresponding to the elements of the domain and range sets, with 1s indicating which pairs are in the relation and 0s indicating which pairs are not. The document provides examples of defining relations from sets and constructing their relation matrices. It also outlines properties that relation matrices must have if the relation is reflexive, symmetric, or antisymmetric.
![RELATION MATRIX
A relation R from a finite set X to a finite set Y can be represented using a
zero-one matrix is called the relation matrix of R.
Let X = {𝑎1, 𝑎2,……, 𝑎𝑚} and Y = {𝑦1, 𝑦1,…......, 𝑦𝑛} be finite set
containing m and n elements, respectively, and R be the relation from X to Y.
Then R can be represented by an m × n matrix 𝑀𝑅 = [mij]mXn, which is
defined as follows:
𝑚𝑖𝑗 =
1,
0,
In the otherwords, the zero-one matrix representing R has 1 as its (i, j) entry xi
is related to yj, and a 0 in this position if xi is not related to yj and a 0 in this
position if xi is not related to yj.
If (𝑥𝑖, 𝑦𝑗) ∈ R
If (𝑥𝑖, 𝑦𝑗) ∈ R](https://image.slidesharecdn.com/matrixrepresentationofarelation-220921170359-da5ee73f/85/MATRIX-REPRESENTATION-OF-A-RELATION-pptx-2-320.jpg)
MATRIX REPRESENTATION OF A RELATION.pptx
- 1. Name: Kiran Kumar Malik Guided by: Dr. Padala Harikrishna Registration Number: 200301120128 Branch: B-Tech in Computer Science and Engineering Section: D Campus: Bhubaneswar
- 2. RELATION MATRIX A relation R from a finite set X to a finite set Y can be represented using a zero-one matrix is called the relation matrix of R. Let X = {𝑎1, 𝑎2,……, 𝑎𝑚} and Y = {𝑦1, 𝑦1,…......, 𝑦𝑛} be finite set containing m and n elements, respectively, and R be the relation from X to Y. Then R can be represented by an m × n matrix 𝑀𝑅 = [mij]mXn, which is defined as follows: 𝑚𝑖𝑗 = 1, 0, In the otherwords, the zero-one matrix representing R has 1 as its (i, j) entry xi is related to yj, and a 0 in this position if xi is not related to yj and a 0 in this position if xi is not related to yj. If (𝑥𝑖, 𝑦𝑗) ∈ R If (𝑥𝑖, 𝑦𝑗) ∈ R
- 3. Example 1: Question: Suppose that A = {1,2,3} AND B = {1,2}.Let R be the relation from A to B containing (a,b) if a ∈ A, b ∈ B and a > b. Solution: R = {(2,1), (3,1), (3,2)} The Matrix Representation is 𝑀𝑅 = 0 0 1 0 1 1
- 4. Example 2: Question: Let A={1, 2, 3, 4}. Find the relation R on A determine by the matrix 𝑀𝑅= 1 0 𝟏 𝟏 0 0 𝟎 𝟏 1 1 𝟎 𝟎 0 0 𝟎 𝟏 Solution: R = {(1,1), (1,3), (2,3), (3,1), (4,1), (4,2), (4,4)}
- 5. PROPERTIES OF A RELATION IN A SET i. If a relation is a reflexive, then all the diagonal entries must be 1. 𝟏 𝟎 𝟎 𝟎 𝟏 𝟎 𝟎 𝟎 𝟏
- 6. ii. If a relation is symmetric, then the relation matrix is symmetric, i.e., mij = mji for every I and j. aij it symmetric element is aji. 𝟏 𝟏 𝟎 𝟏 𝟎 𝟏 𝟎 𝟏 𝟎
- 7. iii. If a relation is antisymmetric, then its matrix is such that if mij = 1 then mji = 0 for I ≠ j. 𝟏 𝟎 𝟏 𝟏 𝟎 𝟎 𝟎 𝟏 𝟏