Prove that every antisymmetric relation is weakly antisymmetric
antisymmetric: if, for all $x,y \in X$, if $xRy$ holds, then $yRx$ does not
weakly antisymmetric: if, for all $x,y \in X$, if $xRy$ and $yRx$ hold, then $x=y$

Question

anonymous · Answer

I don't think anyone currently online can help you ; I recommend that you try 

Math Stack Exchange

anonymous · Answer

thanks lol

anonymous · Answer

@karatechopper can you help?

anonymous · Answer

Proving the contrapositive of this statement might be a little easier to see.

anonymous · Answer

i have a proof from my teacher but i don't understand it.
"Since xRy and yRx never happens, it is weakly antisymmetric"

anonymous · Answer

that's the only thing she wrote

anonymous · Answer

While that is correct, I feel like that is "hand waving" a bit.

anonymous · Answer

If you look at contrapositive of what you are trying to prove, it makes a little more sense.

anonymous · Answer

ok how do i do that?

anonymous · Answer

You are trying to prove, "If the relation R is antisymmetric, then it is weakly antisymmetric."  This is a " P implies Q" type statement.  The contrapostive is where you negate each part of the statement, and switch the order.  So what I am suggesting to prove is this:

"If the relation R is NOT weakly antisymmetric, then it is NOT symmetric".

anonymous · Answer

er...antisymmetric*.  that last word should be antisymmetric.

anonymous · Answer

ok, so if R is not weakly antisymmetric, then xRy and yRx hold would imply x$\neq$y?

anonymous · Answer

That is correct.   Since R is not weakly antisymmetric, there exists an x,y in X such that xRy and yRx hold, but x doenst equal y. Now, it is possible for this relation to be antisymmetric?

anonymous · Answer

no, because xRy AND yRx hold, so this means it's not antisymmetric either.

anonymous · Answer

and thats it!  You have shown that if R is not weakly antisymmetric, then R is not antisymmetric.  Since this is the contrapostive of what we actually wanted to prove, and the contrapositive of a statement is equivalent to the original statement, we are done.  

You have proved that if a relation is antisymmetric, then it is weakly antisymmetric.

anonymous · Answer

ok thanks i understand it now!