Three relations $R_{1}, R_{2}$ , and $R_{3}$ are defined on set A ={ a, b, c} as follows:
(i) $R_{1}$ = { (a, a), (a, b), (a, c), (b, c), (c, a), (b, b), (c, b), (c, c)}
(ii) $R_{2}$ = {( a, b), (b, a), (a, c), (c, a)}
(iii) $R_{3}$ = {(a, b), (b, c), (c, a)}
Discuss each of them from the point of view of being reflexive, symmetric and transitive.

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Solution

$(i) R_{1} =$ Reflexive $: (a, a), (b, b), (c, c) \in R$
Not symmetric $: (a, b) \in R$ but
$(b, a) \notin R_{1}$
Transitive $(a, b) \in R, (b, c) i n R_{1}$
$\Rightarrow (a, c) \in R_{1}$
$(i i) R_{2} -$ Not reflexive $(a, a) \in R L$
symmetric $(a, b), (b, a) \in R_{2}$
$(a, c), (c, a) \in R_{2}$
Not transitive
$(a, b) \in R_{2}, (b, a) \in R_{2}$ but $(a, a) \notin R_{2}$
$(i i i) R_{3}$ Not reflexive $(a, a) \notin R_{3}$
Not symmetric $(a, b) \in R_{3}$
but $(b, a) \notin R_{3}$
Not transitive $(a, b) \in R_{3}$
$(b, c) \in R_{3}$
but $(a, c) \notin R_{3}$

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Similar questions

A = {a, b, c} :

R_{1} = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R_{2} = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)} .

A = {1, 2, 3}

R_{1} = {(1, 1), (1, 3) (3, 1) (2, 2), (2, 1) (3, 3)}

R_{2} = {(2, 2), (3, 1), (1, 3)}

R_{3} = {(1, 3), (3, 3)}

R_{4} = A \times A .

R_{1}, R_{2}, R_{3}, R_{4},

Δ A B C, \frac{b - c}{r_{1}} + \frac{c - a}{r_{2}} + \frac{a - b}{r_{3}} =

\frac{b - c}{r_{1}} + \frac{c - a}{r_{2}} + \frac{a - b}{r_{3}} = 0

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