Antisymmetric relation from a real life

Question

Would you know some examples of antisymmetric relation from a real life? That is, relation $$(x,y)\in R\quad \text{and}\quad (y,x)\in R \rightarrow x=y.$$ Thanks for your help.

Answer 1

On the set of all people set $(x,y) \in R$ if $x$ is the parent of $y$.

Then for any two people $x$ and $y$ it is not possible that $x$ is the parent of $y$ and that $y$ is the parent of $x$. Therefore the implication $$(x,y) \in R \quad\text{and}\quad (y,x) \in R \implies x=y$$ is vacuously true so $R$ is antisymmetric.

Answer 2

The three most common elementary ones in my experience are:

1. Subset relation, like for geometric locus results
2. Real number inequalities
3. Divisibility of positive integers

These are all applicable in real life as long as the mathematical concepts model what you are looking at.

Answer 3

Take the set of all floors from a building. If $f_1$ and $f_2$ are floors, consider$$f_1\mathrel Af_2\quad\text{if}\quad\text{$f_1$ is above than or equal to $f_2$}.$$Then $A$ is an antisymmetric relation.