Question

Consider the following two binary relations on the set $A = \left\{ {a,b,c} \right\}$.
${R_1} = \left\{ {\left( {c,a} \right),\left( {b,b} \right),\left( {a,c} \right),\left( {c,c} \right),\left( {b,c} \right),\left( {a,a} \right)} \right\}$
And
${R_2} = \left\{ {\left( {a,b} \right),\left( {b,a} \right),\left( {c,a} \right),\left( {c,c} \right),\left( {b,b} \right),\left( {a,a} \right),\left( {a,c} \right)} \right\}$
A) ${R_2}$ is symmetric but it is not transitive
B) Both ${R_1}$ and ${R_2}$ are transitive
C) Both ${R_1}$ and ${R_2}$ are not symmetric
D) ${R_1}$ is not symmetric but it is transitive

Answer 1

Hint: Consider the definitions of the both relations carefully. All the elements of both the relations are given. Each option contains the basic properties of relations that are transitivity, symmetry and reflexivity. Use the definitions of the properties and check every option carefully and then eliminate the wrong options.

Complete step-by-step answer:
The given set is $A = \left\{ {a,b,c} \right\}$ .
The relations are defined as:
${R_1} = \left\{ {\left( {c,a} \right),\left( {b,b} \right),\left( {a,c} \right),\left( {c,c} \right),\left( {b,c} \right),\left( {a,a} \right)} \right\}$
And
${R_2} = \left\{ {\left( {a,b} \right),\left( {b,a} \right),\left( {c,a} \right),\left( {c,c} \right),\left( {b,b} \right),\left( {a,a} \right),\left( {a,c} \right)} \right\}$
We will first consider the basic definitions of all the properties including reflexivity, symmetry and transitivity.
For a set $S$ , if we define a relation $R$ then we define the properties as follows:
The relation is said to be reflexive if $\left( {a,a} \right) \in R$ for every element in the relation.
The relation is said to be symmetric if for $\left( {a,b} \right) \in R \Rightarrow \left( {b,a} \right) \in R$ .
The relation is said to be transitive if for $\left( {a,b} \right),\left( {b,c} \right) \in R \Rightarrow \left( {a,c} \right) \in R$ .
We will check the above criteria one by one.
Note that we only need to check symmetry and transitivity as that’s only asked in the options.
Observe that in the relation ${R_1}$ the element $\left( {b,c} \right) \in {R_1}$but $\left( {c,b} \right) \notin {R_1}$ .
Therefore, the relation ${R_1}$ is not symmetric.
On the other hand, if you observe carefully then the relation ${R_2}$ is symmetric.
Similarly, we observe that the relation ${R_1}$ is transitive as for any element of the type $\left( {a,b} \right),\left( {b,c} \right) \in R \Rightarrow \left( {a,c} \right) \in R$ .
Observe that $\left( {b,a} \right),\left( {a,c} \right) \in {R_2}$ but the element $\left( {b,c} \right) \notin {R_2}$ .
Therefore, the element ${R_2}$ is not transitive.
Therefore, we observe that the relation ${R_1}$ is not symmetric but it is transitive.
Similarly, we observe that the relation ${R_2}$ is not transitive but it is symmetric.

Hence, the correct options are A and D.

Note: A relation between two sets is a collection of ordered pairs containing one object from each set. If the object a is from the first set and the object b is from the second set, then the objects are said to be related if the ordered pair (a,b) is in the relation. A function is a type of relation.