Can someone explain antisymmetric versus symmetric relation of sets?

Question

If $$A = \{1,2,3,4\} $$ and $$R = \{(3,3), (4,4), (1,4)\}$$

This example is antisymmetric but not symmetric.

However, the definition of Antisymmetric taken from Merriam-Webster is this:

relating to or being a relation (as “is a subset of”) that implies equality of any two quantities for which it holds in both directions (the relation R is antisymmetric if aRb and bRa implies a = b)

Please forgive my ignorance, but I'm incredibly confused by the definition. I'm obviously wrong, but it suggests to me that there is/should be a (4,1) in R. However, if there was a (4,1) in R that would make R symmetric and not antisymmetric.

I found a previous Question that asked something similar but was answered with the use of Digraphs. It made me understand how to identify antisymmetric relations, but I still don't understand the core concept.

Thank you so much for any help!

Answer 1

You're making a really common mistake, a mistake you probably even know about in abstract terms, but it's hard to recognize it when it shows up in real life. (Especially when the wording is as convoluted as in MW; that's really shoddy work on their part imo). Trying to keep as many words from the original definition as possible, I'll rewrite it so the logic is more clear:

A relation is called antisymmetic when the following statement is true:

• If the relation holds in both directions for two quantities, then the two quantities are equal.

Adding some symbols in there, the bullet point can be rewritten:

• If $(a,b)\in R$ and $(b,a)\in R$, then $a=b$.

Your question amounts to: "I know $(4,1)\in R$, and I know $R$ is antisymmetric, so why isn't $(1,4)\in R$?".

Hopefully the answer is a little more clear now: the bullet point doesn't say anything at all about what happens if all you know is $(a,b)\in R$. (You haven't satisfied both parts of the condition, so the bullet point tells you nothing.)