Anti-symmetric relations

Question

I'm having issues wrapping my head around anti-symmetric examples in specific contexts. I understand that if BOTH $a$, $b$ belong to $\mathbb{R}$ then $a = b$ and if $a \ne b$ then they aren't anti-symmetric.

The context I'm having issues with is inheritance i.e family tree.

$$A = \left\{(x,y) \in P^2 \,\big\vert\, x \text{ is an ancestor of }y \right\}$$

I proved that the relation is NOT symmetric as x can be an ancestor of y but y can't be an ancestor of x. So if it's not symmetric because of the logic reason I gave before it shouldn't be antisymmetric, right?

Answer 1

$$\textit{Anti-symmetry means that mutually related things are identical.}$$

A simple example is the “at most” relation ($\leq$); here, anti-symmetry means $$\text{ if } x \leq y \text { and } y \leq x \text{ then } x = y$$

For the ‘ancestor’ relation, the statements $x \, A \, y$ and $y \, A \, x$ mean that “$x$ is an ancestor of $y$ and $y$ is an ancestor of $x$”, which necessarily means that $x$ and $y$ are the same person.

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On the other hand, $$\textit{Symmetry means that the relation is always mutual.}$$

A simple example is ‘equality’; indeed if $x = y$ then necessarily $y=x$.

Indeed the ancestor relation is not symmetric, as you've reasoned.

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Hope that helps!

Answer 2

A relation $A\subseteq P^2$ (where $P$ is any set) is antisymmetric if, for all $x,y\in P$,

if $(x,y)\in A$ and $(y,x)\in A$, then $x=y$.

The relation $A$ is symmetric if, for all $x,y\in P$,

if $(x,y)\in A$, then $(y,x)\in A$.

For any relation $A$, one (and only one) of the following holds:

• $A$ is symmetric and not antisymmetric;
• $A$ is not symmetric and antisymmetric;
• $A$ is not symmetric and not antisymmetric;
• $A$ is symmetric and antisymmetric.

(Work out an example for each case.) Thus there's no relationship between being symmetric/not symmetric and being antisymmetric/not antisymmetric.

The relation “being an ancestor of” is clearly not symmetric, as you noted. However, it is antisymmetric. Given $x,y\in P$, the statement

if $(x,y)\in A$ and $(y,x)\in A$, then $x=y$

is true, because the statement

$(x,y)\in A$ and $(y,x)\in A$

is false; any statement of the form “if $X$, then $Y$”, where $X$ and $Y$ are arbitrary statement such that $X$ is false, is true.