Determine whether each of the following relations are reflexive, symmetric and transitive:
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}

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Solution

(a) For reflexive, we know that x is divisible by x for all $x ϵ A$ .
$∴ (x, x) ϵ R$ for all $x ϵ R$ . So, R is reflexive.

(b)For symmetry, we know that x is divisible by 2. This means that $(2, 6) ϵ R$ but $(6, 2) d o e s n o t b e l o n g t o R$ . So, R is not symmetric.

(c)For transitiviety, let (x,y)\ belongs R and $(y, z) ϵ R$ , then z is divisible by x.
$\Rightarrow (x, z) ϵ R$
For example, 2 is divisible by 1,4 is divisible by 2
So, 4 is divisible by 1 so, R is transitive.

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