Listing all elements of a set

Question

I was given a question like the following:

Let $A = \Bbb Z$, $B = [-1,\pi]$ , $C=(2,7)$. List all Elements of $A \cap (B^c \cap C)$.

I do not really understand how to got about this problem. I understand $\cap$ means intersection, but I have trouble reading the question; for instance, why place brackets between $B^c \cap C$?

As intersection is an associative operation you could ignore the breaks and have $A\cap B^c\cap C$. — Hirshy, Aug 16 '15 at 17:26
@Hirshy 9 Would i look at B complement as everything that is not in B but intersects C, so like 7 is in B^C and in C? — ChristopherW, Aug 16 '15 at 17:28
Hint: first find the set $D := B^c \cap C$ and then find $A \cap D$. It is a set with a finite number of elements. — molarmass, Aug 16 '15 at 17:28
As $C=(2,7)$ you have $7\notin C$. But yes, $B^c$ is the complement of $B$ in $\mathbb R$. — Hirshy, Aug 16 '15 at 17:30
@molarmass @ Hirshy Like i see how $B^c \cap C$ is everything after $\pi$ to right below 7 but how would i write that? — ChristopherW, Aug 16 '15 at 17:33
$C=(2,7)$ is the set of all real numbers between $2$ und $7$, so if you want to get all real numbers between $\pi$ and $7$ we have...? — Hirshy, Aug 16 '15 at 17:35
What doest that mean? Just take the notation from $C=(2,7)$ and adapt it to $(\pi, 7)$. See also: https://en.wikipedia.org/wiki/Interval_(mathematics)#Notations_for_intervals — Hirshy, Aug 16 '15 at 17:39
@Hirshy Ohhh i did not see that, thank you I understand perfectly — ChristopherW, Aug 16 '15 at 17:40
@coldnumber Damn. How'd you find that? I'm impressed--these questioners must be from the same class or something. Nice find! — Daniel W. Farlow, Aug 16 '15 at 17:45
@DanielW.Farlow Thanks! I watch the elementary-set-theory tag closely. — coldnumber, Aug 16 '15 at 17:49

score 1 · Answer 1 · answered Aug 16 '15 at 17:39

I am going to assume that your domain of discourse is the real numbers $\mathbb{R}$. To that end, structure your computations like so to facilitate an easily produced answer:

$B^c = (-\infty,-1)\cup(\pi,\infty)$
$B^c\cap C=(\pi,7)$
$A\cap(B^c\cap C)=\{4,5,6\}$

This way of going about it is just one way of doing it though. Since $\cap$ is associative, you could computer the intersections of $A$ and $B^c$ first or you could use a variety of other set identities. But the method above is probably the most natural and easiest.

score 1 · Accepted Answer · answered Aug 16 '15 at 17:40

1

We have $A=\mathbb Z, B=[-1,\pi],C=(2,7)$, thus we obtain $B^c=\mathbb R\setminus C=(-\infty,-1)\cup (\pi,\infty)$. As $3<\pi<4$ we have $B^c\cap C=(\pi,7)$. The only integers in $(\pi,7)$ are given by $4,5,6$ so we conclude: $A\cap(B^c\cap C)=\{4,5,6\}$.

On the use of brackets: intersection is an associative operation, so one could ignore the brackets and just write $A\cap B^c\cap C$; as intersection is also commutative we can change the order e.g. look at $A\cap B^c\cap C= B^c\cap A\cap C$. This gives us the opportunity to compute the intersection of two sets that feels the easiest; the brackets given in the question already give you a nice order of computing.

answered Aug 16 '15 at 17:40

Hirshy

5,040

I like how our answers almost identical. One thing though: I always find it odd with these problems how often users fail to state the domain of discourse (which, of course, is absolutely essential). It's like asking how to prove a function is injective, surjective, or bijective without giving the domain or codomain. Rubbish! Haha. – Daniel W. Farlow Aug 16 '15 at 17:44
@DanielW.Farlow Yes, I wasn't sure if I wanted to address that directly and decided to leave it with including it via writing $B^c=\mathbb R\setminus B$...I guess for education purposes I should have included that more clearly, especially as it seems to happen more often (judging from the questions here on MathSE and my experience with new students at university as well). – Hirshy Aug 16 '15 at 17:49
Yes, it's truly bizarre how often it happens, but it suggests something to me--that's a very good sign that a student doesn't truly understand the concept yet. So it can be a good teaching pointer perhaps. Something to keep in mind (I'm about to teach a basic discrete mathematics course so I'll have to keep this in mind for my own practice haha). +1 – Daniel W. Farlow Aug 16 '15 at 17:52
@DanielW.Farlow Good sign indeed, I often try to think of problems students might have with (supposedly easy) questions and I'm still always amazed how many problems I miss because I just couldn't think of them. May I ask where you teach? – Hirshy Aug 16 '15 at 17:58
Sure! I'm going to be a new teacher at Barrie Prep. I'm excited about the discrete class--it's not often you get to teach something like that geared towards high school students. I'm going to have to find a way to pitch it all at the right level though. It's a full year course, and that should give me a fair amount of flexibility. How about you? Sounds like you teach at a university perhaps? – Daniel W. Farlow Aug 16 '15 at 18:01
@DanielW.Farlow (do I have to include this or do you get the notification without them?) actually I'm just a student at RWTH Aachen University, Germany. I'll be finishing my studies to be a teacher for mathematics at a Gymnasium (I guess high school is the english/american equivalent) next February, but will then pursue a PhD in mathematics first instead of being a teacher. My long-term goal is to be a teacher at a university, but as I'm mainly interested in teaching and not in mathematical research, this is unlikely to happen if there are no substantial changes within educational politics. – Hirshy Aug 16 '15 at 18:08
(I get the notifications so long as someone else doesn't comment in I believe--since this is your post, and I'm the only one who has commented) I think we are in very similar situations. I just recently finished college, and my long-term goal is likely to teach at university, but I thought I'd get a feel for teaching in general and then try for grad school in about 3 years or so, depending on how things go. Best of luck with your studies and outmaneuvering education politics! I definitely know that's a huge problem. – Daniel W. Farlow Aug 16 '15 at 18:15
Imo if one gets the feel for teaching, there is no job in this world that gives you more satisfaction. I'm looking forward to next year as while pursuing my PhD I'll already be responsible for various classes at university as assistant to a professor. Best luck to you too! – Hirshy Aug 16 '15 at 18:20

Listing all elements of a set

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