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Standard XII

Mathematics

Transitive Relations

Determine whe...

Question

Determine whether each of the following relations are reflexive, symmetric and transitive:
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}

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Solution

Here, A=N, the set of natural numbers and R={(x,y);y=x+5,x<4}
={(x,x+5): $x ϵ N$ and x < 4}

={(1,6),(2,7),(3,8)}
(a)For reflexive $x, x ϵ R \forall x$ putting $y = x, x \neq y + 5 \Rightarrow (1, 1) / ϵ R$ . So, R is not reflexive.

(b)For symmetrical $(x, y) ϵ R \Rightarrow (y, x) ϵ R$ putting y =x+5, then $x \neq y + 5 \Rightarrow (1, 6) ϵ R$ but $(6, 1) / ϵ R$ . So, R is not symmetric.

(c)For transitivity $(x, y) ϵ R, (y, z) ϵ R \Rightarrow (x, z) ϵ R$ if y =x+5, z=y+5, then $z \neq x + 5$ . Since, $(1, 6) ϵ R$ and there is no order pair in R which has 6 as the first element same in the case for (2,7)and (3,8). So, R is not transitive.

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Determine whether each of the following relations are reflexive, symmetric and transitive: (ii) Relation R in the set N of natural numbers defined as R = {(x, y): y = x + 5 and x < 4}

Determine whether each of the following relations are reflexive, symmetric and transitive:
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}