If R is a symmetric binary relation, what are x and y in set A?

Question

everyone:

I've been reading my textbook for discrete math and a few other textbooks on the topic of binary relations, and finding that I'm struggling to understand the definitions. I think a lot can be clarified for me if someone can answer one thing:

If R is a symmetric binary relation on Set A, what are x and y in A? For me, it's easy to understand what x and y would be if x and y were in R since the elements of a binary relation are ordered pairs (x,y). Can someone help me visualize what they mean when they say x and y are from A?

Thank you so much!

Answer 1

It seems to me that there is some confusion here.

A binary relation (being symmetric or not is not important here) $R$ on a set $A$ is a subset of $A\times A$. Usually, instead of writing $(x,y)\in R$ (and here $x$ and $y$ are elements of $A$), we write $x\mathrel Ry$. Anyway, we only write $x\mathrel Ry$ (or $x\mathrel{\not R}y$ when $(x,y)\notin R$) when $x$ and $y$ are elements of $A$. Otherwise, it doesn't make sense.

Answer 2

$R$ represents a subset of $A\times A$

You asked for a visualisation. The following is an example with $A=\{x,y,z\}$ where $xRx, xRy, yRz, zRy$

Answer 3

One way of visualizing this is just by using an example. Lets think of a set

$R = \{ (x,y)\in \mathbb N^2 |\text{ such that x and y have the same reminder when devided by 3} \}$

($\mathbb N$ represent all non-negative whole numbers $\pm$0)

So for instance, $(3,9),(8,11)\in R , (4,5)\not \in R$. Or using the more common notation: $3R9, 8R11$ but $4\not R 5$

That is a symmetric binary relation on $\mathbb N$. $x$ and $y$ are in $\mathbb N$ (they are whole numbers) and $R$ is a sub set of $\mathbb N ^2 $ (R is composed of ordered pairs of whole numbers) .

Answer 4

It might enlighten you to take a look at the Wikipedia article on the theorem on friends and strangers. Here is the link: https://en.wikipedia.org/wiki/Theorem_on_friends_and_strangers#:~:text=The%20theorem%20says%3A,are%20(pairwise)%20mutual%20acquaintances. The theorem on friends and strangers can be reformulated abstractly as follows: If A is a set of at least 6 elements and R is a symmetric binary relation on A such that no element of A is related (wrt R) to itself, then there exist either a set of 3 mutually-related (wrt R) elements of A, or a set of 3 mutually-unrelated (wrt R) elements of A.