Question

Let ${{R}_{1}}$ be a relation defined by {{R}_{1}}=\left\\{ \left( a,b \right)|a\ge b,a,b\in R \right\\} . Then ${{R}_{1}}$ is
(1) Only reflexive
(2) Both reflexive and transitive
(3) Symmetric, transitive but not reflexive
(4) Neither transitive nor reflexive but symmetric

Answer 1

In this question, we are given the relation in terms of an ordered pair and we have to find out the properties it satisfies. Therefore, we need to understand the definitions of the reflexive, symmetric and transitive properties and check if the given relation satisfies those definitions to obtain the answer to this question.

Complete step-by-step answer:
We are given that the relation R is defined as{{R}_{1}}=\left\\{ \left( a,b \right)|a\ge b,a,b\in R \right\\}..............(1.0). Therefore, it associates a real number a to a real number b if a is greater than or equal to b.
The definition of symmetric, reflexive and transitive properties for a relation between the sets A and B are as follows
1. A relation R from A to B is said to be symmetric when for every ordered pair $\left( a,b \right)$ belonging R, the pair $\left( b,a \right)$ also belongs to R……..(1.1)
2. A relation R from A to B is said to be reflexive if for every element $a\in A$, the ordered pair $\left( a,a \right)$ belongs to R……………..(1.2)
3. A relation R from A to B is said to be transitive when for every ordered pair $\left( a,b \right)$ and $\left( b,c \right)$ belonging to R, the pair $\left( a,c \right)$ also belongs to R……………………..(1.3)
Now, we see that in the given relation{{R}_{1}}=\left\\{ \left( a,b \right)|a\ge b,a,b\in R \right\\}will be symmetric if for every real numbers a and b, $\left( a,b \right)\in {{R}_{1}}\Rightarrow \left( b,a \right)\in {{R}_{1}}$. Thus, it means that $a\ge b\Rightarrow b\ge a$. However, if we take a=5 and b=2, then we see that as $5\ge 2$, $\left( 5,2 \right)\in {{R}_{1}}$. However, as $2\ge 5$ is not true, $\left( 2,5 \right)\notin {{R}_{1}}$. Therefore, using equation (1.1) the relation is not symmetric.
Also, we see that for every real number a, $a\ge a$ always holds. Therefore,$\left( a,a \right)\in {{R}_{1}}$ for every real number a. Therefore, by using equation (1.2), we find that the relation R is also reflexive. For example, as $2\ge 2$ , $\left( 2,2 \right)\in {{R}_{1}}$ .
Now, checking for transitivity, if $\left( a,b \right)\in {{R}_{1}}\text{ and}\left( b,c \right)\in {{R}_{1}}$, then from the definition of the relation
$a\ge b\text{ and }b\ge c...............(1.4)$
we find that a, b and c are real numbers equation (1.4) means that $a\ge b\ge c\Rightarrow a\ge c$. Thus, if $\left( a,b \right)\in {{R}_{1}}\text{ and}\left( b,c \right)\in {{R}_{1}}$ then$\left( a,c \right)\in {{R}_{1}}\text{ }$. Therefore, from the definition of the given relation (equation 1.0), we find that the relation is transitive. For example, as $8\ge 5$ , $(8,5)\in {{R}_{1}}$ and as $5\ge 3$ , $(5,3)\in {{R}_{1}}$ . Now, we see that the relation $8\ge 5$ is also true, therefore$(8,3)\in {{R}_{1}}$ . Thus, $(8,5)\in {{R}_{1}},(5,3)\in {{R}_{1}}\Rightarrow (8,3)\in {{R}_{1}}$ which verifies the transitivity of the relation.
Thus, we find that the relation ${{R}_{1}}$reflexive and transitive. Therefore, option (b) is the correct answer.

Note: We should note that in the equations (1.1), (1.2) and (1.3), there is no restriction that b or c cannot be a. Also, in equation (1.4), the ordering of the variables in the ordered pairs is very important because the ordered pair $\left( a,b \right)$ is not the same as $\left( b,a \right)$ and therefore the pairs should be written in correct order.