Question

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):3x – 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 an 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 1

(i) Relation R in the set A = {1, 2, &#8230;.. , 14) defined as
R = {(x, y) : 3x &#8211; y = 0}
(a) Put y = x, 3x &#8211; x ≠ 0 ⇒ R is not reflexive.
(b) If 3x &#8211; y = 0, then 3x &#8211; x = 0 ⇒ R is not symmetric.
(c) If 3x &#8211; y = 0, 3y &#8211; z = 0, then 3x &#8211; z ≠ 0 ⇒ R is not transitive.
(ii) Relation in tire set N of natural number is defined by
R = {(x, y): y &#8211; x + 5 and x < 4}
(a) Putting y = x, x ≠ x + 5 ⇒ R is not reflexive.
(b) Putting y = x + 5, then x ≠ y + 5 ⇒ R is not symmetric.
(c) If y &#8211; x + 5, z = y + 5, then z ≠ x + 5 ⇒ R is not transitive.
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y): x is divisible by y}
(a) Putting y = x, x is divisible by x ⇒ R is reflexive.
(b) If y is divisible by x, then x is not divisible by y, when x ≠ y ⇒ R is not symmetric.
(c) If x is divisible by y and y is divisible by z, then x is divisible by z,
e.g., 4 is divisible by 2 and 2 is divisible by 1
⇒ 4 is divisible by 1 ⇒ R is transitive.
(iv) Relation R in Z of all integers defined as R = {(x, y): x &#8211; y is an integer}
(a) x &#8211; x = 0 is an integer ⇒ R is reflexive.
(b) x &#8211; y is an integer, so is x ⇒ R is symmetric.
(c) x &#8211; y is an integer, y &#8211; z is an integer and x is also an integer ⇒ R is transitive.
(v) R is a set of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place)
It is reflexive as x works at the same place.
It is symmetric since x and y or y and x work at the same place.
It is transitive since if x, y work at the same place and y, z work at the same place, then x and z also work at the same place.
(b) R: f(x, y): x and y live in the same locality}
With similar reasoning as in part (a), R is reflexive, symmetric and transitive.
(c) R : {(x, y): x is exactly 7 cm taller than y}
It is not reflexive : x cannot be 7 cm taller than x.
It is not symmetric : If x is exactly 7 cm taller than y, then y cannot be exactly 7 cm taller than x.
It is not transitive : If x is exactly 7 cm taller than y and if y is exactly 7 cm taller than z, then x is not exactly 7 cm taller than z.
(d) R = {(x, y): x is wife of y}
R is not reflexive : x cannot be wife of x.
R is not symmetric : x is wife of y but y is not wife of x.
R is not transitive : if x is a wife of y, then y cannot be the wife of anybody else.
(e) R = {(x, y): x is father of y}
It is not reflexive : x cannot be father of himself.
It is not symmetric : x is father of y but y cannot be the father of x.
It is not transitive : If x is father of y and y is father of z, then x cannot be the father of z.