(b) Let A = {5, 6, 7, 8}, B = { 4, 6, 7} and the relations R1 = {(a, b) | a эA, bэB and a > b} R2 = {(a, b) | a э A, b э B and (a – b)2 <=6} (i) Find the sets of ordered pairs in R1, R2 and give their cardinalities |R1|, |R2|. (2 marks) (ii) Draw the directed graphs of R1, R2. (2 marks) (iii) Give the Boolean-matrix representations of R1, R2. (2 marks) (c) Explain what are Reflexive, Symmetric relations using examples. Each relation should contain at least three elements. (2 marks)

Solution.

Part b

i)

$R_1=\{(5,4),(6,4),(7,4),(7,6),(8,4),(8,6),(8,7)\}$

$R_2=\{(a,b)|a\in A, b\in B (a-b)^2<=6\}=\newline\{(5,4),(5,6),(5,7),(6,4),(6,6),(6,7),(7,6),(7,7),(8,6),(8,7)\}$

$|R_1|=7\newline |R_2|=10$

ii)

R1

R2

iii)

$M_{R_1}=\begin{pmatrix} 0&0&0&0&0 \\ 1&0&0&0&0\\ 1&0&0&0&0\\ 1&0&1&0&0\\ 1&0&1&1&0 \end{pmatrix}$

$M_{R_2}=\begin{pmatrix} 0&0&0&0&0 \\ 1&0&1&1&0\\ 1&0&1&1&0\\ 0&0&1&1&0\\ 0&0&1&1&0 \end{pmatrix}$

Part c

Reflexive: A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.

Each element is related to itself.

For a example, R = {(1, 1) (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}

Symmetric: A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a, b ∈ A.

If any one element is related to any other element, then the second element is 

related to the first. 

For a example,

The relation R= {(x, y) ∈ R if and only if $x\neq y$} on the set of all integers is symmetric because $x\neq y$ and $y\neq x.$