# Is this a transitive binary relation?

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

• These seem to be quite mixed up. E.g., the first $R$ is actually transitive (and reflexive is the third) – Hagen von Eitzen Dec 5 '17 at 22:59
• @HagenvonEitzen Oh yikes, I was confident about those. Perhaps I'm misunderstanding what "idA" actually means. Is there anyway you could explain it please? or perhaps a link to a resource that explains it further? (I don't know the name of it, so I'm not too sure what to search for) – user510602 Dec 5 '17 at 23:05
• What context are you approaching this from? Are you going through an axiomatic construction of set theory? The reason I ask is the way relations are defined in a first-order system and the way to actually think about them in terms of what they mean are very different. – David Reed Dec 5 '17 at 23:06
• @DavidReed Honestly I am not sure, sorry. I'm very new to this topic and have only studied the basics, so I'm just approaching it using the methods I've learnt so far (although I don't know the name of them). – user510602 Dec 5 '17 at 23:11
• @Vlart Ok. If you could tell me what brought you to this topic, like what your goal is in terms of using this topic once you understand it, it would help me in determining the best way to answer. – David Reed Dec 5 '17 at 23:14

## 1 Answer

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.

• Wow, really informative - thank you so much! Although I would like to finish this current book as I'm following a set track ('syllabus' is perhaps the correct English word?). Do you happen know the correct ordering of the matches in my post? I feel lost trying to understand where 'id' comes from, as it isn't mentioned anywhere in the question, unlike 'A'. – user510602 Dec 6 '17 at 0:44
• @ user510602 id is either the equality relation ''=" or the identity function. It looks like its probably the identity function....f(x) = x – David Reed Dec 6 '17 at 3:50
• @user 510602 can you provide me with the name of the book you're using? – David Reed Dec 6 '17 at 3:57