Question

Which of the following relations is only symmetric?

• A. “less than equal to” on a set of Real numbers
• B. “is a multiple of ” on the set of positive integers
• C. “is perpendicular to” on a set of a coplanar lines
• D. “is the father of ” on a set of family members.

Answer 1

The correct option is C

“is perpendicular to” on a set of a coplanar lines

Explanation for the correct option:

The correct option is C:“is perpendicular to” on a set of a coplanar lines

This relation “is perpendicular to” on a set of a coplanar lines is not reflexive as a line cannot be
perpendicular to itself.

if $l_1$ is perpendicular to $l_2$ then $l_2$ is also perpendicular to $l_1$,therefore given relation is symmetric
$l_1\perp l_2$and $l_2\perp l_3$does not implies $l_1\perp l_3$

so given relation is not transitive.

Explanation for incorrect options:

Option A: “less than equal to” on a set of Real numbers

Let $a,b,c\in A$ where $A$ is a set of real numbers.
Then $R=\left\{(a,b):a\le b,a,b\in A\right\}$ is :

Reflexive:$a\le a\Rightarrow (a,a)\in R$ (Yes)

Symmetric:$a\le b\Rightarrow (a,b)\in R$, but

$a\le b\Rightarrow a<bora=b$

$\Rightarrow b=abutb⪇aso(b,a)\notin R$(No)

Transitive: $(a,b)\in R\Rightarrow a\le b,(b,c)\in R\Rightarrow b\le c$

$\Rightarrow a\le c,\left(a,c\right)\in R$

Hence , The relation is not symmetric

Option B: “is a multiple of ” on the set of positive integers

Let R = set of positive integers. Then,
$R=\left\{(a,b):\mathrm{a}\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}\mathrm{b},a,b\in R\right\}$
Reflexive: Every positive integer is a multiple itself, so,
$(a,a)\in R$ (Yes)
Symmetric: $(a,b)\in R$ ⇒$a$ is a multiple of $b$ but $b$ is a not multiple of $a$ $\Rightarrow (b,a)\notin R$ (No)
Transitive :$(a,b)\in R\Rightarrow$ $a$ is a multiple of $b$
$\Rightarrow a=mb\forall m\in N$.
$(b,c)\in R\Rightarrow b$is a multiple of $c\Rightarrow b=nc\forall n\in N$
$\therefore a=m\times nc\Rightarrow a=mnc\forall m,n\in N$
$\Rightarrow a$ is a multiple of$c\Rightarrow (a,c)\in R.$ (yes)

Hence , the relation is not symmetric

Option D: “is the father of ” on a set of family members.

A person cannot be his own father, so relation is not reflexive.
If a is father of b, then b cannot be father of a, so relation is not symmetric.
If a is father of b, b is father of c, then a cannot be father of c, so relation is not transitive.
Hence ,the option(C) is the correct answer.