Question

Let R be any relation in the set A of human beings in a town at a particular time. If $R=\left\{ \left( x,y \right):x\,is\,exactly\,7\,cm\,taller\,than\,y \right\}$, Then R is
(a) Not symmetric
(b) Reflexive
(c) Symmetric but not transitive
(d) An equivalent relation

Answer 1

Hint: In this question, we will check conditions of reflexive, symmetric and transitive relation separately and use it to find the conclusion.

Complete step-by-step solution -
In given question, A is set of, all human beings in a town and relation R is given as:
$R=\left\{ \left( x,y \right):x\,is\,exactly\,7\,cm\,tall\,then\,y \right\}$.
Now, any relation is reflexive, if for all elements in a set, an element is related to itself, that is, for all x belongs to A, $\left( x,x \right)$ belongs to R.
Now, in a given relation, $\left( x,x \right)$ belongs to R means that x is 7 cm taller than y. But this cannot be true, as no person can be taller than himself. Therefore, R is not reflexive.
Also, in any relation, if x is related to y such that then y is also related to x, then R is symmetric. That is, if $\left( x,y \right)$ belongs to R such that $\left( y,x \right)$ also belongs to R, then R is symmetric.
Now, in a given relation, $\left( x,y \right)$ belongs to R means that x is exactly 7 cm taller than y. Now, if x is taller than y, then y is 7 cm shorter than x. But $\left( y,x \right)$ belongs to R which means y is 7 cm taller than x, which is not true. So, $\left( y,x \right)$ does not belong to R.
Therefore, R is not symmetric.
Also, in any relation, if x is related to y and y is related to z, such that, then x is related to y, then relation is transitive. That is, if $\left( x,y \right)$ and $\left( y,z \right)$ belong to R, then R is transitive.
Now, in given relation,
$\left( x,y \right)$ and $\left( y,z \right)$ belongs to R means that x is 7 cm taller than y and y is 7 cm taller than z. the, x will be 7+7=14 cm taller than z. But $\left( x,z \right)$ belongs to R will mean x is exactly 7 cm taller than z, which is not true. So, $\left( x,z \right)$ does not belong to R.
Therefore, R is not transitive.
Hence the correct option answer is option (a).

Note: In this type of question, where writing tabular form of a set is not possible, we consider examples of the situation given to solve the question. We proceed to a solution with a definition of reflexive, symmetric and transitive and check whether the given relation follows the definition or not.