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I need to prove or disprove, that R is antisymmetric. This is my set: $$ R=\{(1,1),(1,2), (1,4), (2,1), (2,2), (3,2), (3,3), (4,4)\} $$

I proved that it is not antisymmetric in the following manner: $$ \text{Definition of antisymmetric relation}\\ (x,y)\in R \land (y,x)\in R \implies x=y\\ (x,y)\in R \land x\not = y \implies (y,x) \notin R\\ \text{I found a example in set where this does not apply}\\ (2,1)\in R \land 2\not = 1 \implies (1,2) \notin R\\ \text{This is false, so that means that the relation is not antisymmetric.} $$

This means thet this set is not antisymmetric??

Right??

Thanks!!!

depecheSoul
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2 Answers2

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You would have to look of pairs of the form $(a, b)$ where $a \neq b$. Then look for the pair of the form $(b, a)$. If it exists, then you have $(a, a)$ and $(b, b)$ in your system.

The point is you can only have one or the other, hence antisymmetric.

Don Larynx
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Your reasoning is right but you need to write it a bit more clearly. Something like the following should be fine.

Suppose $R$ is anti-symmetric. We see that $(2,1)\in R$ and $2\neq 1$ and so by anti-symmetry we should have that $(1,2)\notin R$. However, this is not the case because $(1,2)\in R$ and so we have reached a contradiction. It follows that the assumption that $R$ is anti-symmetric is false.

I should probably add, getting used to using symbols when writing proofs is good... but honestly, most people prefer reading proper English sentences in proofs. You should get into the habit of writing your math out in prose, sure with the occasional symbol where needed, so that you can actually read it out loud and it makes sense. It also shows that you actually understand what you're writing, and you're not just trying to remember a lot of rules about symbol manipulation.

Dan Rust
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