AntiSymmetric help

Question

So I understand Symmetric =

(a,b) > (b,a) so e.g. A = {(1,2), (2,1) = Symmetric

And Anti symmetric = {(1,2), (1,1) = Anti symmetric since the (1,2) is present but no (2,1).

But what if we have {(2,1), (1,1,)}

Does That constitute as Anti symmetric? Or does the first integer have to be less than the second integer? i.e. (1,2), (2,1).

Sorry if it's a bit confusing but my question basically is, for anti-symmetric what do we define as (a,b)

Thanks

Answer 1

Does That constitute as Anti symmetric? Or does the first integer have to be less than the second integer? i.e. (1,2), (2,1).

Ah! Now I understand your question.

No; the magnitude of the operands doesn't matter. The members of the pairs do not have to be ordered by size. They don't even have to be orderable.

It's just that if any pair $(a,b)$ is in an antisymmetric relation then it follows that either $a=b$ or that the complementary pair $(b,a)$ is not is not in the relation, exclusively.

Conversely, if there is any case of non-identical $a$ and $b$, where both $(a,b)$ and $(b,a)$ are in the relation then it follows that the relation is not antisymmetrical.

So if $(1,2)$ is in an antisymmetrical relation then $(2,1)$ cannot be. Likewise if $(2,1)$ is in an antisymmetrical relation then $(1,2)$ cannot be. While if both are in a relation then it is not antisymmetrical.

Answer 2

Symmetry of a relation $R$ on a set $S$ is defined by $$(a,b) \in R \Rightarrow (b,a) \in R \quad \forall a,b\in S$$ Anti-symmetry is defined by $$(a,b)\in R \Rightarrow (b,a) \notin R \quad\forall a\ne b\in S\tag2$$ (thanks to @Graham)
This clearly qualifies $\{(2,1),(1,1)\}$ as an antisymmetric relation on $\{1,2\}$ or any superset thereof.

Answer 3

Given a relation $R$ over the set $A$.

The relation is symmetric if $\forall a,b\in A: [(a,b)\in R\implies (b,a)\in R]$

That is to say: If $(a,b)$ is in a symmetric relation then so is $(b,a)$.

The relation is antisymmetric if $\forall a,b\in A: [(a,b),(b,a)\in R \implies a=b]$

That is to say, if both $(a,b)$ and $(b,a)$ are in an antisymmetric relation, then they are identical. (Though the converse need not hold.)

So if $A=\{1,2,3\}$ then $\{(1,1),(2,2), (1,3)\}$ is an example of an antisymmetric relation, while $\{(1,2), (2,1), (3,3)\}$ is an example of a symmetric relation.

Additionally: $\{(1,1),(2,2),(3,3)\}$ is an example of a relation that is both symmetric and antisymmetric.

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Now, consider the relation over the integers, $R_{_{(>)}}=\{(a,b): \forall a,b\in\mathbb{Z}, a>b\}$

Since $\forall a,b\in\mathbb{Z}, a > b \implies b\not> a$ then $R_{_{(>)}}$ is not symmetric.

However, since $\not\exists a,b\in\mathbb{Z}: a>b \land b>a$ then $R_{_{(>)}}$ is actually antisymetric, though vaccuously.

(Remember that $(P\Rightarrow C)$ is equivalent to $(\neg P \lor C)$, so if the premise is never true then the implication always holds.)