Need a lil help to get started

[misterfister]

Let A = {v, w, x, y, z}. Determine the number of relations on A that are reflexive and symmetric.

not so sure where to start

 

[HallsofIvy]

Do you know what "reflexive" and "symmetric" mean? Start by writing down the definitions of those words.
Do you see why there are a total of \(\displaystyle 2^5= 32\) possible relations (including those that are reflexive and symmetric.)?

 

[misterfister]

Do you know what "reflexive" and "symmetric" mean? Start by writing down the definitions of those words.
Do you see why there are a total of \(\displaystyle 2^5= 32\) possible relations (including those that are reflexive and symmetric.)?

Hmm why is there 32? How do you break down v, w, x, y, z into different relations?

I was thinking rearrange them in every possible way but wouldn't that be 5!?

 

[daon2]

Hmm why is there 32? How do you break down v, w, x, y, z into different relations?

I was thinking rearrange them in every possible way but wouldn't that be 5!?

No, the number of arrangements is \(\displaystyle 5! = 120\) but that is not relevant.

A relation on a set A is a subset of AxA. So the number of relations on A is the number of subsets of AxA, which is \(\displaystyle 2^{|A\times A|}=2^{25}.\)

A relation \(\displaystyle R\subseteq A\times A\) is symmetric if \(\displaystyle (x,y)\in R \implies (y,x)\in R\).

A relation \(\displaystyle R\subseteq A\times A\) is reflexive if \(\displaystyle (x,x)\in R\) for all \(\displaystyle x\in A\).