Could anyone please chack my task on equivalence relation? THank you!! In this task it says: if $f\colon X \to Y$ is a transformation. We define the relation $R$ to $X$ :
$x'\sim_R x'' \Leftrightarrow f (x') = f(x'')$
Show if it´s equivalence relation.
reflexive: yes x´~ R x´´ <> f(x´) =f (x´´) 1 ~2 <> 4 =4
symmetrical, no
x´~ R x´´ <> f(x´) = f (x´´) it´s not clearly defined to which x the image belongs
transitive, yes x´~ R x´´ <> f(x´) = f (x´´) 1 ~ 2 <> 4 = 4 x´´~ Rx´´´<> f(x´´) = f(x´´´) 2 ~ 3 <> 4 =4 then x´~ R x´´´<> =f(x´) = f(x´´´) 1 ~ 3 <> 4 =4
another one Ihave to solve.
There is an amount consisting of three elements $X= \{a,b,c\}$ Give three relations of X with the characteristics:
1) reflexive, not symmetrical, not transitive
2) symmetrical, not reflexive, not transitive
3) transitive, not reflexive, not symmetrical.
Thank you. I would be happy about both, explanation and a way to solve these tasks. Sophia
For the first question, note that you are saying that $x'$ and $x''$ are related if and only if they have the same image under $f$.
That keyword same is a tell-tale sign that you have an equivalence relation. Just try to spell out the three properties, such as for each $x$, is it true that $x$ and $x$ have the same image under $f$, and you'll see what I mean.
(By the way, you should see soon in your course that every equivalence relation arises this way.)
For the rest, just play around a bit - this is an exercise from which you can learn something only by trying to do it yourself. Starting with the reflexive/not reflexive part is probably sensible. Show us what you have tried, and you'll get help.