Question

For any two real numbers $\theta$ and $\phi$, we define , if and only if ${\sec ^2}\theta - {\tan ^2}\phi = 1$. The relation R is
(a) Reflexive but not transitive
(b) Symmetric but not reflexive
(c) Both reflexive and symmetric but not transitive
(d) An equivalence relation

Answer 1

We need to understand the relation given in the question and use the properties of relation. A relation consists of sets of ordered pairs of elements satisfying the relation.

Complete step by step solution:
According to the question, we are given two real numbers $\theta$and $\phi$, which are related under the relation R such that the elements are $\theta R\phi$. Now the equation ${\sec ^2}\theta - {\tan ^2}\phi = 1$will hold true only when $\theta = \phi$. So, if ${\sec ^2}\theta - {\tan ^2}\phi = 1$
\Rightarrow \theta = \phi \\\ \Rightarrow R \\\
is Reflexive and Symmetric.
Since, there are only two elements given in the question hence, R cannot be transitive.
For a relation to be transitive we need a minimum of three elements, say a,b and c such that if $aRb$ and $bRc$ hold true then $aRc$ must hold true as well. But in this question we are only dealing with $aRa$ or $bRb$ and $aRb$ or $bRa$. Hence the relation R is Reflexive and Symmetric.

Note: We are only considering two elements and two relations. Had there been another element say $\alpha \in C$ , such that, $\theta = \phi = \alpha$ , such that $aRb$ and $bRc$ holds true so that$aRc$ is also true, then the relation R is transitive.When a relation R is reflexive, symmetric and transitive then the relation R will be an equivalence relation.