The following argument claims to prove that the requirement that an equivalence relation be reflexive is redundant. In other words, it claims to show that if a relation is symmetric and transitive, then it is reflexive. Find the mistake in the argument.
“Proof: Let Rbe a relation on a set A and suppose R is symmetric and transitive. For any two elements x and y in A, if xRy then yRx since R is symmetric. Thusit follows by transitivity thatx R x, and hence R is reflexive.”
The mistake in the given argument.
Let R be a relation on a set A and suppose R is symmetric and transitive. For any two elements x and y in A, if x R y then y R x since R is symmetric. But then it follows by transitivity that x R x. Hence R is reflexive.”
It could happen that for some there does not exist such that x R y. in this case, we cannot rely on the argument provided in the exercise prompt to show that R is transitive.
Consider the set and the relation
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