Is Every Asymmetric Relation Also Antisymmetric?

[icantadd]

I ran across the following statement, in a tutorial on logic,

If R is an asymmetric relation then R is antisymmetric.

Perhaps, the above is true. I will attempt to argue it is not. Okay, then suppose R really is asymmetric, for example the relation (<) So for an arbitrary (c,d) c < d then d >= c. From this we should get R is antisymmetric. It R is antisymmetric though, ( c < d and d < c then d = c )

Take for example the natural numbers, suppose 1. 1 < x and x < 1 means 1 = x. Which would mean that 1 is less than 1, and 1 is greater than 1, which poses contradiction because 1 is not less than 1.

 

[n_bourbaki]

c<d and d<c certainly does not imply c=d. It is impossible for c<d and d<c to be simultaneously satisfied.

 

[HallsofIvy]

I ran across the following statement, in a tutorial on logic,

If R is an asymmetric relation then R is antisymmetric.

Perhaps, the above is true. I will attempt to argue it is not. Okay, then suppose R really is asymmetric, for example the relation (<) So for an arbitrary (c,d) c < d then d >= c. From this we should get R is antisymmetric. It R is antisymmetric though, ( c < d and d < c then d = c )

Take for example the natural numbers, suppose 1. 1 < x and x < 1 means 1 = x. Which would mean that 1 is less than 1, and 1 is greater than 1, which poses contradiction because 1 is not less than 1.

The first thing you should do is state the definitions, as given in that tutorial, of "asymmetric" and "antisymmetric".