Proof for relation

[nwicole]

What is wrong with the following argument, which supposedly shows that any relation R on X that is symmetric and transitive is reflexive.

Argument: Let x in X. Using symmetry, we have ( x, y ) and ( y, x ) both in R. Since ( x, y ), ( y, x ) are in R by transitivity, we have ( x, x ) in R. Therefore R is reflexive.

thank you so much

 

[pka]

What is wrong with the following argument, which supposedly shows that any relation R on X that is symmetric and transitive is reflexive.

Argument: Let x in X. Using symmetry, we have ( x, y ) and ( y, x ) both in R. Since ( x, y ), ( y, x ) are in R by transitivity, we have ( x, x ) in R. Therefore R is reflexive.

This is an old favorite. The mistake is in assuming that the domain of R is all of X.

 

[HallsofIvy]

For example, the relation on X= {1, 2, 3} defined by {(2, 2), (3, 3), (2, 3), (3, 2)} is both symmetric and transitive but not reflexive because it does not contain (1, 1).