Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) R = {(x, y) : x and y work at the same place}
(ii) R = {(x, y) : x and y live in the same locality}
(iii) R = {(x, y) : x is wife of y}
(iv) R = {(x, y) : x is father of and y}

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Solution

(i) Reflexivity:
$Let x be an arbitrary element of R . Then, x \in R \Rightarrow x and x work at the same place is true since they are the same . \Rightarrow (x, x) \in R So, R is a reflexive relation .$

Symmetry:
$Let (x, y) \in R \Rightarrow x and y work at the same place \Rightarrow y and x work at the same place \Rightarrow (y, x) \in R So, R is a symmetric relation .$

Transitivity:
$Let (x, y) \in R and (y, z) \in R . Then, x and y work at the same place . y and z also work at the same place . \Rightarrow x, y and z all work at the same place . \Rightarrow x and z work at the same place . \Rightarrow (x, z) \in R So, R is a transitive relation .$

(ii) Reflexivity:
$Let x be an arbitrary element of R . Then, x \in R \Rightarrow x and x live in the same locality is true since they are the same . So, R is a reflexive relation .$

Symmetry:
$Let (x, y) \in R \Rightarrow x and y live in the same locality \Rightarrow y and x live in the same locality \Rightarrow (y, x) \in R So, R is a symmetric relation .$

Transitivity:
$Let (x, y) \in R and (y, z) \in R . Then, x and y live in the same locality and y and z live in the same locality \Rightarrow x, y and z all live in the same locality \Rightarrow x and z live in the same locality \Rightarrow (x, z) \in R So, R is a transitive relation .$

(iii)
Reflexivity:
$Let x be an element of R . Then, x is wife of x cannot be true . \Rightarrow (x, x) \notin R So, R is not a reflexive relation .$

Symmetry:
$Let (x, y) \in R \Rightarrow x is wife of y \Rightarrow x is female and y is male \Rightarrow y cannot be wife of x as y is husband of x \Rightarrow (y, x) \notin R So, R is not a symmetric relation .$

Transitivity:
$Let (x, y) \in R, but (y, z) \notin R Since x is wife of y, but y cannot be the wife of z, y is husband of x . \Rightarrow x is not the wife of z \Rightarrow (x, z) \in R So, R is a transitive relation .$

(iv)
Reflexivity:
$Let x be an arbitrary element of R . Then, x is father of x cannot be true since no one can be father of himself . So, R is not a reflexive relation .$

Symmetry:
$Let (x, y) \in R \Rightarrow x is father of y \Rightarrow y is son / daughter of x \Rightarrow (y, x) \notin R So, R is not a symmetric relation .$

Transitivity:
$Let (x, y) \in R and (y, z) \in R . Then, x is father of y and y is father of z \Rightarrow x is grandfather of z \Rightarrow (x, z) \notin R So, R is not a transitive relation .$

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Similar questions

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

Q. 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 }

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 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}