Question

Let L denote the set of all straight lines in a plane. Let a relation R be defined by $\alpha R\beta \Leftrightarrow \alpha \bot \beta ,\alpha ,\beta \in L$ . Then R is:
(A). Reflexive
(B). Symmetric
(C). Transitive
(D). None of these

Answer 1

Hint: First observe the relation given in question. Find all the properties related to the relation. Try to substitute values or variables to check their properties. Try to use definitions of Reflexive, transitive, symmetric, anti symmetry. First and the foremost thing to do is use the definition of relation and say that the given relation satisfies all the conditions needed for it to be a relation. Next see the range on which relation is defined. While checking properties, substitute few elements only which satisfy the range given in the question

Complete step-by-step solution -
Relation:- The relations in maths are also called as binary relation over se x,y is a subset of the Cartesian product of the set x,y that is the relation contains the elements of set XxY. XxY, is a set of ordered pairs consisting of element x in X and y in Y. It encodes the information, $\left( x,y \right)$ will be in relation set if and only if x is related y in the relation property given.
i). Reflexive relation:- A relation R is said to be reflexive, if a is an element in the range. If the pair $\left( a,a \right)$ belongs to relation then relation is reflexive. In other words, if a is related to itself by given relation then relation is reflexive.
ii). Symmetric Relation:- A relation R is said to symmetric, if a,b are elements in the range. If the pair $\left( a,b \right)$ belongs to relation then $\left( b,a \right)$ must belong to relation. In other words if a is related to b, then b must also be related to a for a relation to be symmetric.
iii). Transitive Relation:- If $\left( a,b \right)$ belongs to relation and $\left( b,c \right)$ also belongs to the relation, then $\left( a,c \right)$ must belong to relation to make it transitive.
Anti symmetric Relation:- If $\left( a,b \right)$ and $\left( b,a \right)$ are given belong to relation, then a must be equal to b to make the relation as anti symmetric. If a relation is reflexive, Transitive and Symmetric, then it is an equivalence relation
Set L denotes all the straight lines possible.
The relation R is given in the question as 2 lines are in relation if 1st line is perpendicular to the 2nd line.
By definition trying the different types, we get:
a). Reflexive: we know that a line is coincident to itself but not perpendicular. So, we can say the relation “not reflexive”
b). Transitive: If given A is perpendicular to B and B is perpendicular to C. By basic knowledge of geometry it is possible that A is parallel to C. So, it is not transitive.
c). Symmetric: If A is perpendicular to B then B is also perpendicular to A. So, it is symmetric. The relation R is symmetric.
Therefore option (B) is correct.

Note: If you are not able to say directly. Try to find an example to prove it is not that relation type. Here for reflexive we take x-axis is not perpendicular to x and here for transitive we take positive x-axis perpendicular to y and y perpendicular to negative x-axis but positive, negative x-axis not perpendicular.