-Self studying 2Yr Undergraduate Discrete Math
-Background: Calculus 3, probability 1, statistics 1, linear algebra 1, ODEs 1, stochastics 1
Hi, I'm studying relations and want to confirm I understand binary relations. I learned that a relation $aRb$ (let's say the relation is a is a child of b) is the Cartesian product $A\times B$, fair enough. So mentally I draw two circles, one for set $A$, one for set $B$, and I can put a pair $(a,b)$ in a third circle $A \times B$ if the $a$ is indeed a child of $b$.
The definition of a binary relation I have here is:
"A binary relation R on A is a relation between A and A, and a binary relation can always be composed with itself. Its inverse is a binary relation on the same set."
I tend to use visuals to help me understand, but here is where my drawing breaks down. Using the child-parent relationship again, I have a circle $A$ and I can move two a's $(a_1, a_2)$ if $a_1$ is indeed the child of $a_2$, right? The problem with this example is I can't see the significance/specialty of a binary relation.
REMOVED - DISREGARD BOTTOM HALF OF QUESTION
Second part to my question: it goes on to define a transitive closure $$ R^* = R^0 \times R^1 \times\dots R^n $$ as well as positive transitive closure $$ R^+ = R^1 \times R^2 \times\dots R^n $$ I don't understand the $R^0$ relation; I read that it's the domain of $R$ (also called an identity relation) but going back to my visual example, isn't the domain of my circle $A$ just $A$? So much like multiplying by $1$ or multiplying a matrix by $I$, is this just a fancy way to say essentially nothing? What are some applications of these two things $R^*, R^+ $?
Sorry for the long question. Pls ELI5 since I'm self studying, I don't have a teacher for this stuff :)