How do the Properties of Relations work?

This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions.

There are five properties for a relation:

Reflexive - $R \rightarrow R$

Symmetrical - $R \rightarrow S$ ; $S \rightarrow R$

Antisymmetrical - $R \rightarrow S$ && ($R \rightarrow R$|| $S \rightarrow S$)

Asymmetrical -$R \rightarrow S$ && !($R \rightarrow R$|| $S \rightarrow S$)

Transitive - if $R \rightarrow S$ && $S \rightarrow T$, then $R \rightarrow T$

If that's not what you call the properties in English, then please let me know- I have to study it in German, unfortunately, and these are my rough translations.

Continuing on, I just don't know what to do with this information practically. The examples of the book are horrible:

1) "Is the same age as" is apparently reflexive, symmetrical and transitive. 2) "Is related to" is also apparently reflexive, symmetrical and transitive. 3) "Is older than" is asymmetric, antisymmetric and transitive.

There are more useless examples like this. I have no idea how it comes to these conclusions because we're talking about a literal statement. I was hoping perhaps for some real mathematical examples, but the book falls short on those.

I would greatly appreciate it if somebody could explain the above example and perhaps give me a better use for Relations other than... that. Also, how can a relation be a- and antisymmetrical at the same time? Don't they cancel each other out?

I'd like to change the notation of your definitions, since $R$, $S$ and $T$ would usually be used to stand for the relations themselves (and $x, y$ and $z$ would be more commonly chosen for the objects that might bear the relation to each other).

Reflexive - For all $x: xRx$

Example reflexive relation: $xRy$ stands for '$x$ is a factor of $y$' (in the set of natural numbers)

Symmetric - For all $x,y$: if $xRy$ then $yRx$

Example symmetric relation: $xRy$ stands for '$x$ and $y$ are $2$ metres apart' (in the set of all people in a particular room)

Antisymmetric - For all $x,y$: if $xRy$ and $yRx$ then $x = y$

Example antisymmetric relation: $xRy$ stands for '$x$ is a factor of $y$' (in the set of natural numbers)

Asymmetric - For all $x,y$: if $xRy$ then not $yRx$

Example asymmetric relation: $xRy$ stands for '$x$ is taller than $y$' (in the set of all people)

Transitive - For all $x,y,z$: if $xRy$ and $yRz$ then $xRz$

Example transitive relation: $xRy$ stands for '$x$ is taller than $y$' (in the set of all people)

I think you're thinking about this in the wrong way. Properties don't "work", properties are things that are true for the given relation.

For instance, you say that the a relation has the reflexive property if it satisfies the condition that all elements are related to themselves. Similarly, it has the symmetric property if for all a and b, if a is related to b then b is related to a.

You could simply say that "∀a : a R a" each time, but because this is something that happens often, this particular property has been given a name, i.e. reflexivity.

Certain types of specific relations are also given names. For instance, a partial order is reflexive, antisymmetric and transitive, and equivalence relations are reflexive, symmetric and transitive. Again, these are just names that aren't strictly needed, but they make it more convenient talking about these kinds of things.