Proving Subsets: A Venn Diagram Approach

leilei
Messages
8
Reaction score
0
Proof subset?

Given three sets A, B, and C, set X = (A-B) U (B-C) U (C-A) and
Y = (A∩B∩C) complement C. Prove that X is subset of Y. Is Y necessarily a subset of X? If yes, prove it. If no, why?
---When I draw the two venn diagrams X and Y, they are the same, but I don't know how to prove it...

Can someone help me out here...
Thanks in advance!
 
Physics news on Phys.org
The usual proof for such a statement is: Let x be an element from X and try to prove that it is also an element of Y. So if x is in X, you know that it is in A but not in B, and/or it is in B but not in C, and/or it is in C but not in A. You want to show that it must be in (A∩B∩C)C. If it is in (A∩B∩C) then it would be in A and B and C, so it being in the complement means it is at least not in A or not in B or not in C. So you could suppose it is both in A and B and show that it cannot be in C.

So let me write this out in the right order:
Let x \in X. Suppose that x \in A, x \in B. Then definitely, x \not\in A - B, x \not\in C - A, because if it is in A it cannot be in any set from which we remove all elements of A (and similarly for B). But it must be in one or more of (A-B), (B-C) and (C-A), so it must be in (B - C). That is, x is in B (which we knew) but not in C. So if x is not in C, it cannot be in the intersection of C with whatever set you make up. In particular, it is not in A \cap B \cap C. Therefore, it must be in the complement of that set, which is called Y.

Now try to do the same reasoning for Y \subset X. You have already shown by your Venn diagram that if x lies in Y, it must lie in X. So try to prove it in the same way as I just did.
 
Thanks a lot !
 
There is, however, a technical problem with "Let x \in X"- the proof collapses is X is empty. Far better to start "IF x \in X". That way, if X is empty, the hypothesis is false and the theorem is trivially true.
 
You are right, obviously the empty set is a subset of any set S (vacuously, all its elements are also in S).
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Meaning of "identify" & variations in different math context
Replies
5
Views
500
MHB Subsets of permutation group: Properties
Replies
18
Views
2K
I Implication, Boolean Expression and Venn Diagrams
Replies
12
Views
7K
I Trouble with a passage in Zariski & Samuel's Commutative algebra text
Replies
5
Views
623
I Proving Convexity of the Set X = {(x, y) E R^2; ax + by <= c} in R^2
Replies
5
Views
2K
I Meaning of "passage to the quotient"?
Replies
3
Views
439
Proving Completeness property of ##\mathbb{R}## using Dedekind cuts
Replies
20
Views
3K
MHB Operations on Sets: Proving A⊆B⊆C & A∪B=B∩C
Replies
2
Views
2K
I Proving SL_2(C) Homeomorphic to SU(2)xT & Simple Connectedness
Replies
1
Views
2K
I Finitely Generated Modules and Their Submodules ....
Replies
13
Views
2K

Hot Threads

Recent Insights

Back
Top