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Write down the properties of relations in details?

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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→R

Symmetrical - R→S ; S→R

Antisymmetrical - R→S && (R→R|| S→S)

Asymmetrical -R→S && !(R→R|| S→S)

Transitive - if R→S && S→T, then R→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?
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