I am trying to understand binary relations, and am therefore trying to match up binary relation R on A to its property.
I feel my major difficulty is recognizing transitive properties, so if someone please clarify whether I am correct, I would be very grateful:
$R ◦ R ⊆ R$ is reflexive
$R = R^{-1}$ is irreflexive
$\operatorname{id}_A ⊆ R$ is symmetric
$\operatorname{id}_A ∩ R = ∅$ is anti-symmetric
$R ∩ R^{-1} ⊆ \operatorname{id}_A$ is transitive
I'm getting the hunch in speaking with you that you are not going through an axiomatic construction of set theory. If that's true there's really no need to try and approach relations in the way that you are. Here's how to think of transitive relations:
It's best to start with an example. Think of the integers, $\mathbb{Z} = \{\ldots -3,-2,-1,0,1,2,3,\ldots\}$
An example of a binary relation on $\mathbb{Z}$ is the relation $\,\leq\,$.
Note that for any integers $x,y,z\ $,
$$\text{if }\ x \leq y,\text{ and } y \leq z\ \text{ then } x \leq z \\$$
A relation that has this property is said to be transitive. Another relation we can put on the integers
(in fact any set) is the subset relation "$\subseteq$". This is also a transitive relation, since
$$\text{if }\ A \subseteq B,\text{ and } B \subseteq C\ \text{ then } A \subseteq C \\$$
Formally, a relation $R$ on a set $A$ is a subset of $A \times A$. This is the approach your book is taking. It is
unnecessary for your purposes..and confusing. I would advise getting another book, a good one for
what you are after is "Mathematical Thinking:Problem Solving and Proofs". I will touch on it
anyways though.
$A \times A$ is the set of all ordered pairs $ (x,y) $ where $x,y \in A$. Given an arbitrary relation $R$:
$$ xRy \iff (x,y) \in R$$
For instance, going back to our first example, the relation $\leq $ on $\mathbb{Z}$ is the set of all ordered pairs
$(x,y) \in \mathbb{Z} \times \mathbb{Z}$ such that $x \leq y$
Again nobody actually thinks of relations this way except in one specific context.