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:
- "Is the same age as" is apparently reflexive, symmetrical and transitive.
- "Is related to" is also apparently reflexive, symmetrical and transitive.
- "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 asymmetrical and antisymmetrical at the same time? Don't they cancel each other out?