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**MATHS**

**Question 1:**

Determine whether each of the following relations are reflexive, symmetric and transitive:

(i)Relation R in the set *A *= {1, 2, 3…13, 14} defined as

R = {(*x*, *y*): 3*x *− *y *= 0}

(ii) Relation R in the set **N **of natural numbers defined as

R = {(*x*, *y*): *y *= *x *+ 5 and *x *< 4}

(iii) Relation R in the set *A *= {1, 2, 3, 4, 5, 6} as

R = {(*x*, *y*): *y *is divisible by *x*}

(iv) Relation R in the set **Z **of all integers defined as

R = {(*x*, *y*): *x *− *y *is as integer}

(v) Relation R in the set *A *of human beings in a town at a particular time given by

(a) R = {(*x*, *y*): *x *and *y *work at the same place}

(b) R = {(*x*, *y*): *x *and *y *live in the same locality}

(c) R = {(*x*, *y*): *x *is exactly 7 cm taller than *y*}

(d) R = {(*x*, *y*): *x *is wife of *y*}

(e) R = {(*x*, *y*): *x *is father of *y*}

Answer

(i) *A *= {1, 2, 3 … 13, 14}

R = {(*x*, *y*): 3*x *− *y *= 0}

∴R = {(1, 3), (2, 6), (3, 9), (4, 12)}

R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R.

Also, R is not symmetric as (1, 3) ∈R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0]

Also, R is not transitive as (1, 3), (3, 9) ∈R, but (1, 9) ∉ R.

[3(1) − 9 ≠ 0]

Hence, R is neither reflexive, nor symmetric, nor transitive.

(ii) R = {(*x*, *y*): *y *= *x *+ 5 and *x *< 4} = {(1, 6), (2, 7), (3, 8)}

It is seen that (1, 1) ∉ R.

∴R is not reflexive.

(1, 6) ∈R

But,

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