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Is it true that every relation which is symmetric and transitive is also reflexive ?

Last updated date: 20th Jun 2024
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Hint-Make use of the definitions of reflexive , symmetric, transitive functions and solve this.

A relation R on a set A is called reflexive if (a,a) $\in$ R holds for every element a $\in$ A .i.e. if set A = {a,b} then R = {(a,a), (b,b)} is reflexive relation.
A relation R on a set A is called symmetric if (b,a) $\in$ R holds when (a,b) $\in$ R.i.e. The relation R={(4,5),(5,4),(6,5),(5,6)} on set A={4,5,6} is symmetric
A relation R on a set A is called transitive if (a,b) $\in$ R and (b,c) $\in$ R then (a,c) $\in$ R for all a,b,c $\in$ A.i.e. Relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.
So, from these statements we can say that the given statement is false, Let us try to prove this, Let us prove it by taking a counter example
For example , take a relation R ={(1,1,),(1,2),(2,1),(2,2)} on A={1,2,3}
So, from this example we can clearly infer from the definition that the given relation is symmetric since(1,2),(2,1) $\in$R and transitive since (1,2)(2,1) $\in$R and also (1,1) $\in$R but not reflexive since
(3,3) $\notin$ R
So, from this we can write that the statement is given is false

Note: Only when the given statement is false, we can prove it by taking a counterexample else if the given statement is true , then we cannot solve it by taking a counter example.