Mathematics solution 1
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Mathematics solution EXERCISE 1.1

Explanation of Relations

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) : 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}

(i) Relation R in the set A = {1, 2, 3, ..., 13, 14}

R = {(x, y) : 3x – y = 0}

  • Reflexive: No
  • Symmetric: Yes
  • Transitive: Yes

Explanation:

The relation R is not reflexive because for every element x in the set A, (x, x) is not in R since 3x - x = 2x ≠ 0.

It is symmetric because if (x, y) is in R, then 3x - y = 0, which means y = 3x. Thus, (y, x) is also in R.

It is transitive because if (x, y) and (y, z) are in R, then 3x - y = 0 and 3y - z = 0, which implies 3x - z = 3x - 3y + 3y - z = 0. So, (x, z) is also in R.

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

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

  • Reflexive: No
  • Symmetric: No
  • Transitive: No

Explanation:

The relation R is not reflexive because for every element x in the set N, (x, x) is not in R since y = x + 5 ≠ x.

It is not symmetric because if (x, y) is in R, then y = x + 5. However, (y, x) will not be in R because it violates the condition x < 4.

It is not transitive because if (x, y) and (y, z) are in R, then y = x + 5 and z = y + 5. However, it does not imply that z = x + 10.

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

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

  • Reflexive: Yes
  • Symmetric: No
  • Transitive: No

Explanation:

The relation R is reflexive because for every element x in the set A, (x, x) is in R since x is divisible by x.

It is not symmetric because if (x, y) is in R, where y is divisible by x, it does not imply that (y, x) is in R.

It is not transitive because if (x, y) and (y, z) are in R, where y is divisible by x and z is divisible by y, it does not imply that z is divisible by x.

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

R = {(x, y) : x – y is an integer}

  • Reflexive: Yes
  • Symmetric: Yes
  • Transitive: Yes

Explanation:

The relation R is reflexive because for every integer x in the set Z, (x, x) is in R since x - x = 0, which is an integer.

It is symmetric because if (x, y) is in R, then x - y is an integer. Thus, (y, x) is also in R since y - x = -(x - y), which is also an integer.

It is transitive because if (x, y) and (y, z) are in R, then x - y and y - z are integers. Thus, (x, z) is also in R since x - z = (x - y) + (y - z), which is an integer.

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

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

  • Reflexive: Yes
  • Symmetric: Yes
  • Transitive: Yes

Explanation:

The relation R is reflexive because every person works at the same place as themselves.

It is symmetric because if x and y work at the same place, then y and x also work at the same place.

It is transitive because if x and y work at the same place and y and z work at the same place, then x and z also work at the same place.

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

  • Reflexive: Yes
  • Symmetric: Yes
  • Transitive: Yes

Explanation:

The relation R is reflexive because every person lives in the same locality as themselves.

It is symmetric because if x and y live in the same locality, then y and x also live in the same locality.

It is transitive because if x and y live in the same locality and y and z live in the same locality, then x and z also live in the same locality.

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

  • Reflexive: No
  • Symmetric: No
  • Transitive: No

Explanation:

The relation R is not reflexive because a person cannot be exactly 7 cm taller than themselves.

It is not symmetric because if x is exactly 7 cm taller than y, it does not imply that y is exactly 7 cm taller than x.

It is not transitive because if x is exactly 7 cm taller than y and y is exactly 7 cm taller than z, it does not imply that x is exactly 7 cm taller than z.

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

  • Reflexive: No
  • Symmetric: No
  • Transitive: No

Explanation:

The relation R is not reflexive because a person cannot be their own wife.

It is not symmetric because if x is the wife of y, it does not imply that y is the wife of x.

It is not transitive because if x is the wife of y and y is the wife of z, it does not imply that x is the wife of z.

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

  • Reflexive: No
  • Symmetric: No
  • Transitive: No

Explanation:

The relation R is not reflexive because a person cannot be their own father.

It is not symmetric because if x is the father of y, it does not imply that y is the father of x.

It is not transitive because if x is the father of y and y is the father of z, it does not imply that x is the father of z.