Question

If A = $\left\{ 1,2,3,4 \right\}$ , define relations on A which have properties of being symmetric but neither reflexive nor transitive.

Answer 1

Hint: First we are going to look at the definition of symmetric, reflexive and transitive. And after that we will define a relation which satisfies that it is symmetric but neither reflexive nor transitive. And that set will be the final answer.

Complete step-by-step answer:
Let’s start our solution by first writing all the definition of the terms:
Symmetric: If we have a set containing two elements ‘a’ and ‘b’, then if the relation set has (a,b) then it must have (b,a) then we can say that it is symmetric.
Reflexive: If we have a set containing two elements ‘a’ and ‘b’, then if the relation set has (a,a) and (b,b) then we can say it is reflexive.
Transitive: If we have a set containing three elements ‘a’ , ‘b’, and ‘c’ then if the relation set has (a,b) and (b,c) then it must have (a,c) for transitive.
Now we have stated all the required definitions.
Now we will create a relation set which satisfies symmetric but neither reflexive nor transitive.
R =$\left\{ \left( 1,2 \right),\left( 2,1 \right) \right\}$
We can see that it satisfies the above condition.

Note: One can see that we can create many relation sets which satisfies that it must be symmetric but neither reflexive nor transitive in A. The example that we have given is just one of them.