Question

Consider the below relational matrix (MR) on set A = {1, 2, 3, 4}. Identify if MR is reflexive,
symmetric, antisymmetirc, transitive or none of those. Justify your answers with proper explanation.

Answer 1

Answer:

1 1 1 0

0 1 0 0 = MR is the answer

0 0 1 1

1 1 0 1

Explanation:

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Answer 2

Given :

A = {1, 2, 3, 4}

$M_R = \left[\begin{array}{cccc}1&1&1&0\\0&1&0&0\\0&0&1&1\\1&0&0&1\end{array}\right]$

To find :

Whether MR is reflexive ,  symmetric , antisymmetric , transitive or none of those.

Solution :

$M_R = \left[\begin{array}{ccccc}&1&2&3&4\\1&1&1&1&0\\2&0&1&0&0\\3&0&0&1&1\\4&1&0&0&1\end{array}\right]$

We get R = {(1 , 1) , (1 , 2) , (1 , 3) , (2 , 2) , (3 , 3) , (3 , 4) , (4 , 1) , (4 , 4)}

Reflexive relation : A relation R on a set A is said to be reflexive if each x ∈ A , (x , x) ∈ R.

(1 , 1 ) , (2 , 2) , (3 , 3) and (4 , 4) ∈ R

Therefore, reflexive

Symmetric relation : A relation R on a set A is said to be symmetric if (x , y) ∈ R then (y , x) ∈ R for all x , y ∈ A

(1 , 2) ∈ R but (2 , 1) ∉ R

(1 , 3) ∈ R but (3 , 1) ∉ R

(3 , 4) ∈ R but (4 , 3) ∉ R

(4 , 1) ∈ R but (1 , 4) ∉ R

Therefore, not symmetric

Antisymmetric relation : A relation R on a set A is said to be antisymmetric if xRy and yRx ⇒ x = y for all x,y ∈ A

(1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) ∈ R

Therefore, antisymmetric

Transitive relation : A relation R on a set A is said to be transitive if xRy and yRz ⇒ xRz for all x , y , z ∈ A

Since, no such relation is present.

Therefore, not transitive.