Consider the proposition: :et R be a relation on a set X. If R is symmetric and transitive, then it is reflexive. The proposition is false. However, someone wrote the following proof. What is wrong with it.
Proof: "Let x and y be elements of X. If xRy, then yRx since R is symmetric. Then, by transitivity, we have that xRx, so that R is reflexive. QEd."
My answer (wrong): I said that in the proposition, it says: "If R symmetric AND transitive",
but in the proof, it is : "if R symmetric, then transitive".
so in the proof, it is "then" instead of "and".
Can someone explain to me what else is wrong with my answer. thank you!!
What if R is the empty relation?
More completely: your proof assumes that there is always x and y which are related in the first place, and that's where the error lies. If there is some element unrelated to any other element, then you cannot apply this proof which assumes the existence of a y related to x. However symmetry and transitivity do not give any kind of existence to you.