Q4 Write down any 4*4 matrix having only zeros and ones. (3*5=15) a) Draw the Directed graph of that matrix. b) List the ordered pairs in the relation on set {1, 2, 3, 4} corresponding to this matrix. c) Determine whether the relations on this graph/ matrix are Reflexive, Symmetric and Anti-symmetric. d) Determine whether the relation for this graph is equivalence or not?

Let the matrix is $A = \begin{bmatrix} 1 & 0 & 0 &0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0& 1\\ \end{bmatrix}$ .

a) Directed graph is as follows:

b) Ordered pairs in the relation on set {1, 2, 3, 4} corresponding to this matrix = $\{ (1,1),(2,2),(3,3),(4,4)\}$

c) Given matrix is Reflexive since $A_{ii} = 1 \ forall \ i = 1,2,3,4$ .

Symmetric since $A_{ij} = A_{ji}$ for all $i,j = 1,2,3,4$

and Anti-symmetric since $A_{ij} = A_{ji}$ for all $i,j = 1,2,3,4$

d) Given matrix is transitive also since $A_{ii} = 1 , A_{ij} = 0 \ if \ i \neq j$ .

Hence the relation for this graph is equivalence.