Relations Problem | Discrete Mathematics

[JamesBennettBeta]

I can't understand how to solve this one. I search for solutions everywhere and I couldn't get any. Can someone explain me this problem?

 

[tkhunny]

How do inverses work?

An inverse function (if it exists) works like this...
[math]If\;f(x) = y,\;then f^{-1}(y) = x.[/math]
Is it necessarily different for an inverse relation?

 

[Dr.Peterson]

First let's fix the typos and try to correct the grammar! Here's what I'm guessing it's supposed to say:

If [MATH]\sigma[/MATH] is a relation from A to B, then define the relation [MATH]\sigma^{-1}[/MATH] to be its inverse.​

(i) Let A = {1, 2, 3} and B = {a, b, c}. The relation [MATH]\sigma[/MATH] from A to B is defined as {(1, b), (3, a)}. Find [MATH]\sigma^{-1}[/MATH].​

(ii) The relation [MATH]\rho[/MATH] is defined as [MATH]\rho = \{(a,b)\in\mathbb{Z}\times\mathbb{Z}: a - b\text{ is divisible by }3\}[/MATH]. Show that [MATH]\rho[/MATH] is an equivalence relation.​

These are two quite different problems, so we'll have to discuss them separately. (I don't see why the first line is there at all, or why (ii) is related to it.) Most important, we'll need to know how much you do understand, in order to help.

(i) What is the definition of an inverse relation? Since this relation is defined directly as a set of ordered pairs, your answer will also be a set of ordered pairs. What will they be?

(ii) What is the definition of an equivalence relation? What three things do you have to prove? Please show the parts you have done, and where you are stuck.

 

[pka]

I can't understand how to solve this one. I search for solutions everywhere and I couldn't get any. Can someone explain me this problem?

Comment: To James, if you had not posted the image of the question then I would not have believed that anyone could have had nerve enough to have written such a question.
The first, i. is absolutely trivial: two ordered pairs.
But the second is moderately advanced.
ii. If \(\displaystyle \sigma\subseteq A\times A\) then
\(\displaystyle \sigma\) is an equivalence relation on \(\displaystyle A\iff~1)\;\Delta_A=\{(x,x) : x\in A\}\subset \sigma,\; 2)\; \sigma^{-1}=\sigma~\&~\;3)\;\sigma\circ\sigma\subseteq\sigma\)