Question

Three relations $ {R_1} $ , $ {R_2} $ and $ {R_3} $ are defined on set $ A = \left\{ {a,b,c} \right\} $ as follows:

I. $ {R_1} = \left\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {a,c} \right),\left( {b,c} \right),\left( {c,a} \right),\left( {b,b} \right),\left( {c,b} \right),\left( {c,c} \right)} \right\} $
II. $ {R_2} = \left\{ {\left( {a,a} \right),\left( {b,a} \right),\left( {a,c} \right),\left( {c,a} \right)} \right\} $
III. $ {R_3} = \left\{ {\left( {a,b} \right),\left( {b,c} \right),\left( {c,a} \right)} \right\} $

Discuss each of them from the point of view of being reflexive, symmetric and transitive.

Answer 1

Hint: For each problem to find reflexive need to have $ \left( {a,a} \right) $ that is same type of elements should be there and the condition to find whether the set is symmetric is that if $ a $ is related to $ b $ and $ b $ is related to $ a $ . If $ a $ related to $ b $ and $ b $ related to $ c $ then $ a $ related to $ c $ is the condition for transitive.

Complete step-by-step answer:
The set A is $ A = \left\{ {a,b,c} \right\} $ .
 (i)
Given
 $ {R_1} = \left\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {a,c} \right),\left( {b,c} \right),\left( {c,a} \right),\left( {b,b} \right),\left( {c,b} \right),\left( {c,c} \right)} \right\} $ .

Reflexive:
To find the given set is reflexive then if $ \left( {x,x} \right) $ is in $ {R_1} $ where $ x $ is in $ \left\{ {a,b,c} \right\} $ .
We know that $ \left( {a,a} \right) $ , $ \left( {b,b} \right) $ and $ \left( {c,c} \right) $ is in $ {R_1} $ .
Hence, it is clear that $ {R_1} $ is reflexive.

Symmetric:
To find the given set is symmetric then if $ \left( {x,y} \right) $ is in $ {R_1} $ then $ \left( {y,x} \right) $ should be in $ {R_1} $ where $ y,x $ belongs to set $ \left\{ {a,b,c} \right\} $ .
It is known that $ \left( {a,b} \right) $ is in $ {R_1} $ but $ \left( {b,a} \right) $ is not in $ {R_1} $ .
Hence, we can say that $ {R_1} $ is not symmetric.

Transitive:
To find the given set is transitive then if $ \left( {x,y} \right) $ is in $ {R_1} $ and $ \left( {y,z} \right) $ in $ {R_1} $ then $ \left( {x,z} \right) $ is in $ {R_1} $ where $ x,y \in \left\{ {a,b,c} \right\} $ .
Here, we know $ \left( {a,b} \right) $ is in $ {R_1} $ and $ \left( {b,c} \right) $ is in $ {R_1} $ and also $ \left( {a,c} \right) $ is also in $ {R_1} $ .
Hence, $ {R_1} $ is transitive.

(ii)
Given $ {R_2} = \left\{ {\left( {a,a} \right),\left( {b,a} \right),\left( {a,c} \right),\left( {c,a} \right)} \right\} $ .

Reflexive:
To find the given set is reflexive then if $ \left( {x,x} \right) $ is in $ {R_2} $ where $ x $ is in $ \left\{ {a,b,c} \right\} $ .
Given $ \left( {a,a} \right) $ is in $ {R_2} $ .
But $ \left( {b,b} \right) $ and $ \left( {c,c} \right) $ are not in $ {R_2} $
Hence it is clear that $ {R_2} $ is not reflexive.

Symmetric:
To find the given set is symmetric then if $ \left( {x,y} \right) $ is in $ {R_2} $ then $ \left( {y,x} \right) $ should be in $ {R_2} $ where $ y,x $ belongs to set $ \left\{ {a,b,c} \right\} $ .
It is known that $ \left( {b,a} \right) $ is in $ {R_2} $ .
But $ \left( {a,b} \right) $ is not in $ {R_2} $ .
Hence $ {R_2} $ is not symmetric.

Transitive:
To find the given set is transitive then if $ \left( {x,y} \right) $ is in $ {R_2} $ and $ \left( {y,z} \right) $ in $ {R_2} $ then $ \left( {x,z} \right) $ is in $ {R_2} $ where $ x,y \in \left\{ {a,b,c} \right\} $ .
Here $ \left( {b,a} \right) $ is in $ {R_2} $ and $ \left( {a,c} \right) $ is in $ {R_2} $ and also $ \left( {b,c} \right) $ is not in $ {R_2} $ .
Hence, $ {R_2} $ is not transitive.

(iii)
Given $ {R_3} = \left\{ {\left( {a,b} \right),\left( {b,c} \right),\left( {c,a} \right)} \right\} $
Reflexive:
To find the given set is reflexive then if $ \left( {x,x} \right) $ is in $ {R_3} $ where $ x $ is in $ \left\{ {a,b,c} \right\} $ .
Given $ \left( {a,a} \right) $ is not in $ {R_3} $ .
Hence it is clear that $ {R_3} $ is not reflexive.

Symmetric:
To find the given set is symmetric then if $ \left( {x,y} \right) $ is in $ {R_3} $ then $ \left( {y,x} \right) $ should be in $ {R_3} $ where $ y,x $ belongs to set $ \left\{ {a,b,c} \right\} $ .
It is known that $ \left( {a,b} \right) $ is in $ {R_3} $ .
But $ \left( {b,a} \right) $ is in $ {R_3} $ .
Hence $ {R_3} $ is not symmetric.

Transitive:
To find the given set is transitive then if $ \left( {x,y} \right) $ is in $ {R_3} $ and $ \left( {y,z} \right) $ in $ {R_3} $ then $ \left( {x,z} \right) $ is in $ {R_3} $ where $ x,y \in \left\{ {a,b,c} \right\} $ .
Here $ \left( {a,b} \right) $ is in $ {R_3} $ and $ \left( {b,c} \right) $ is in $ {R_3} $ and also $ \left( {a,c} \right) $ is not in $ {R_1} $ .
Hence, $ {R_3} $ is not transitive.

Note: Equivalence relation is expressed as the relation among elements of a particular set that could be transitive, reflexive or symmetric. To prove any equivalence relation, first we have to prove that it is reflexive relation, symmetric relation and transitive relation.