Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Can anyone give me some hints about this homework? I`m really stuck. Thank you.

Let R is a binary relation in A. Define T and S using R, such that S is the smallest symmetric relation which $R \subseteq S$ and T is the biggest symmetric relation which $T \subseteq R$. (In other words, S and T are binary relations, defined using the sets R,$\varnothing, Id_{A}, A \times$ A and the operations $\cup,\cap , \ ^{-1}, \circ $)

share|improve this question
3  
@DidierPiau: First line: "Can anyone give me some hints about this homework?" –  Arturo Magidin Jan 31 '12 at 20:17
1  
@DidierPiau: The second paragraph: describing/defining the smallest symmetric relation on $A$ that contains $R$ using $R$, $\varnothing$, $\mathrm{Id}_A$, $A\times A$, $\cap$, $\cup$, ${}^{-1}$, and $\circ$; and describing the largest symmetric relation on $A$ that is contained in $R$ using the same ingredients... –  Arturo Magidin Jan 31 '12 at 20:30
1  
@Arturo: ...and asking no question at all, in the end. Sure, reading your post, I see that you assume the question is: Show that such an $S$ and such a $T$ exist. This interpretation makes sense, of course, but it is NOT in the original post. (To raise the quality of the formulation of the questions on the site seems a worthy goal, I would say. Don't you think?) –  Did Jan 31 '12 at 21:16
1  
@DidierPiau: I don't follow your comment. The "imperative" in the second paragraph is "Define $T$ and $S$... using blah". So I'm not assuming that the question is "Show that such an $S$ and $T$ exist", but rather that the question is "Can you help me do this problem?" And the problem is "Use blah to define/construct S and T such that..." This is clearer than many a post we have received, though certainly not the best possible. But my point is that the post does contain an explicit question about an explicit problem. –  Arturo Magidin Jan 31 '12 at 21:35
1  
@Didier: Yes, I think better questions are a worthy goal. But I honestly don't understand your particular objection to this one (as opposed to others). The way I read this question, the OP quoted a problem from his assignment/book (the second paragraph), and asked for help with that problem (the first paragraph). A common occurrence here. You seem to be reading it as if the second paragraph is not a well-defined problem, I'm reading it as something I would see in a textbook, not as an attempt by the OP at framing a task. –  Arturo Magidin Feb 1 '12 at 1:56

2 Answers 2

up vote 3 down vote accepted

For $T$, first ask yourself what ordered pairs in $R$ absolutely need to be removed to make a symmetric relation; for $S$, what ordered pairs in absolutely have to be added to $R$ to make a symmetric relation. The problem then is to use the allowable tools to remove them.

  1. What kind of relation $P$ on $A$ has the property that $P=P^{-1}$?

  2. If $P$ is a relation on $A$, what does $P\cap P^{-1}$ look like? Answering this should help you with $T$.

  3. Ask and answer a question similar to (2) that helps with $S$.

There is an alternative way to get at $S$. It’s often the case that the smallest whatsit containing a given object is the intersection of all of the whatsits containing it; this works whenever the intersection of whatsits is again a whatsit. Can you see why it works? (Note that this is a pretty common construction, so it’s well worth understanding.)

In this case you want $S$ to be the smallest symmetric relation on $A$ containing $R$, so in your problem a whatsit is a symmetric relation on $A$. Answering the following questions should help you with $S$.

  1. Is it true that the intersection of symmetric relations on $A$ is symmetric?

  2. Is there at least one symmetric relation on $A$ that contains $R$?

share|improve this answer

General Hint. Can you describe the property "$U$ is a symmetric relation" using $\cap$, $\cup$, ${}^{-1}$, and/or $\circ$? Note that "$U$ is symmetric" if and only if $(a,b)\in U\Longleftrightarrow (b,a)\in U$ and that $(x,y)\in U\Longleftrightarrow (y,x)\in U^{-1}$.

Hints for $S$. If $S_1$ and $S_2$ are symmetric relations, and both contain $R$, is $S_1\cap S_2$ a symmetric relation that contains $R$? Is there at least one symmetric relation that contains $R$ (not necessarily the smallest one...)

Hints for $T$. What should you remove from $R$ to get something symmetric? What should you keep?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.