Transformation or relation 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.