I\'m leaning towards (2). But I\'m still unsure. Could you explain to me which o

Question

I'm leaning towards (2). But I'm still unsure. Could you explain to me which of the three are correct, and why?

For the following statement S, either (1) S is true and the proof is correct, (2) S is true and the proof is incorrect, (3) S is false and the proof is incorrect. Explain which of these occurs.


S: Every symmetric and transitive relation on a nonempty set is an equivalence relation.

Proof Let R be a symmetric and transitive relation defined on a nonempty set A. We need only show that R is reflexive. Let x be an element in A. We will show that x R x. Let y be an element of A such that x R y. Since R is symmetric, y R x. Now x R y. and y R x. Since R is transitive, x R x. Thus R is reflexive.

Explanation / Answer

I'm leaning towards (2). But I'm still unsure. Could you explain to me which of the three are correct, and why?

For the following statement S, either (1) S is true and the proof is correct, (2) S is true and the proof is incorrect, (3) S is false and the proof is incorrect. Explain which of these occurs.


S: Every symmetric and transitive relation on a nonempty set is an equivalence relation.
Proof Let R be a symmetric and transitive relation defined on a nonempty set A. We need only show that R is reflexive. Let x be an element in A. We will show that x R x. Let y be an element of A such that x R y. Since R is symmetric, y R x. Now x R y. and y R x. Since R is transitive, x R x. Thus R is reflexive. (3) S is false and the proof is incorrect. FOR A RELATION TO BE REFLEXIVE,IF X IS ANY ELEMENT OF THE SET A , THEN [X,X] SHALL BE AN ELEMENT OF A X A IN OTHER WORDS EVEN IF THERE IS ONE ELEMENT Z IN A AND WE DO NOT HAVE [Z,Z] AS AN ELEMENT OF A X A ..THEN THE RELATION IS NOT REFLEXIVE... THE ABOVE PROOF STARTS WITH AN ELEMENT [X,Y] IN THE SET A X A AND GOES ON TO PROVE THAT IN SUCH A CASE [X,X] IS ALSO PRESENT IN A X A . BUT WHAT HAPPENS IF AN ELEMENT Z PRESENT IN A , BUT DOES NOT HAVE SYMMETRIC RELATION WITH ANY OTHER ELEMENT IN A? IN SUCH A CASE THE ABOVE PROOF FAILS AND WE DO NOT HAVE [Z,Z] AS AN ELEMENT OF A X A AND HENCE THE RELATION IS NOT REFLEXIVE... SO THE KEY DIFFERENCE IN DEFINITION ...TO BE NOTED HERE IS REFLEXIVE PROPERTY SHALL HOLD GOOD FOR EVERY ELEMENT IN A.. THAT IS IF A=[1,2,3] , WE SHALL HAVE [(1,1),(2,2),(3,3)] IN THE RELATION FOR IT TO BE CALLED REFLEXIVE ... WHERE AS SYMMETRIC AND TRANSITIVE PROPERTY SHALL HOLD GOOD FOR WHATEVER ELEMENTS PRESENT IN A X A ...THAT IN THE BOVE CASE THE RELATION IS SYMMETRIC IF A X A HAS [(1,2),(2,1)]... IT NEED NOT FEATURE ELEMENT 3 AT ALL!!

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