Find the error in the following argument by providing a coun

Find the error in the following argument by providing a counterexample. \"The reflexive property is redundant in the axioms for an equivalence relation. If x ~ y, then y ~ x by the symmetric property. Using the transitive property, we can deduce that x ~ x.\" Also, find out where exactly the flaw is in the argument.

Solution

The problem is that, given x, there may be no y with x y. For example, for any set X, consider the empty relation where x y is never true. This is symmetric and transitive, but not reflexive.

For example, let A={1,2,3} and let:

={(1,1),(1,2),(2,1),(2,2)}

This relation is symmetric, and transitive, but it is not reflexive, since it does not contain (3,3)-in fact, NO element of A is related to 3.