MHB Undirected graph proof, Set intersection

AI Thread Summary
The discussion centers on formal proofs related to set theory and graph theory. In Part A, the user attempts to prove that the intersection of two sets A and B, each larger than n/2, is nonempty by contradiction, concluding that assuming an empty intersection leads to a contradiction regarding the union of the sets. In Part B, the user seeks guidance on proving that an undirected graph with each vertex having a degree greater than x/2 is connected, with hints suggesting that demonstrating a diameter of at most 2 would suffice. Clarifications are provided on terminology and the relationship between the sets and their cardinalities. Ultimately, the discussion emphasizes the importance of understanding the connections between the two parts of the proof.
c4nn3t
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Hello all, I'm a bit stumped when it comes to formal proofs. I

PART A: "Let A,B ⊆{1,2...n} be two sets with A,B > n/2. Prove that the intersection of A ∩ B is nonempty."
This part I used contradiction, but didn't get everything. I assumed that if the intersection of A and B was empty, then A∪B is A+B, which is n>2 + n>2 = n. A∪B⊆{1,2,...n}, so therefore A∪B must be <= |{1,2...n}| = n. Then, n >= |A∪B| > n, which results in a contradiction.
Is this enough of a formal proof, or is there a step I missed?PART B: "Let Z=(J,K) be an undirected graph on x vertices. Each vertex has degree >x/2. Prove that Z is connected."
I wasn't sure how to start off proving this one. Supposedly solving part A helps solve this. Any two nodes are adjacent to more than >x/2 of the nodes, but I'm not sure how to turn this into a formal proof.

I'd really appreciate some input to get going, thanks so much in advance!
 
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c4nn3t said:
Hello all, I'm a bit stumped when it comes to formal proofs. I

PART A: "Let A,B ⊆{1,2...n} be two sets with A,B > n/2. Prove that the intersection of A ∩ B is nonempty."
This part I used contradiction, but didn't get everything. I assumed that if the intersection of A and B was empty, then A∪B is A+B, which is n>2 + n>2 = n. A∪B⊆{1,2,...n}, so therefore A∪B must be <= |{1,2...n}| = n. Then, n >= |A∪B| > n, which results in a contradiction.
Is this enough of a formal proof, or is there a step I missed?PART B: "Let Z=(J,K) be an undirected graph on x vertices. Each vertex has degree >x/2. Prove that Z is connected."
I wasn't sure how to start off proving this one. Supposedly solving part A helps solve this. Any two nodes are adjacent to more than >x/2 of the nodes, but I'm not sure how to turn this into a formal proof.

I'd really appreciate some input to get going, thanks so much in advance!

Hi c4nn3t, :)

The idea behind your first proof is correct. What you have used is the fact that for any two finite sets $A,\,B\subseteq \{1,\,2,\,\cdots,\,n\}$,

\[|A\cup B|=|A|+|B|-|A\cap B|\]

Supposing $|A\cap B|=0$ we have,

\[|A\cup B|=|A|+|B|>n\]

Also since, $A\cup B\subseteq \{1,\,2,\,\cdots,\,n\}\Rightarrow |A\cup B|\leq n$

Both of these conditions cannot be true.
 
c4nn3t said:
PART B: "Let Z=(J,K) be an undirected graph on x vertices. Each vertex has degree >x/2. Prove that Z is connected."
Hint: Prove that not only the graph is connected, but its diameter is $\le 2$. Here the diameter is the greatest distance between any pair of vertices, and the distance between two vertices is the length of a shortest path between them. In other words, diameter 2 means that one can go from one vertex to any other vertex in at most two steps.
 
AH, okay, now that first part makes sense, thanks so much for that!

I'm not still sure if I've grasped the second part. Also, I forgot to mention, J is the number of vertices, and E is the number of edges in the graph, I'll update the original post to reflect that. So I'd start out saying for a graph Z, each vertex J has degree greater than x/2. This must mean that K is also > x/2. I want to show that K is <= 2 between any two vertices, which I think is the step I'm having difficulty finding.
So once I prove that the graph is diameter <=2, how does that lead to it being connected? Ultimately, I state that the G is connected if J has connectivity n-1. Do I need to consider if the graph is Eulerian?

Still quite confused with this one, maybe I'm not picking at my brain hard enough... (Fubar)
 
c4nn3t said:
Also, I forgot to mention, J is the number of vertices, and E is the number of edges in the graph
But post #1 says that $J$ is the set of vertices, and $K$ (and not $E$) is the set of edges. Meanwhile, the number if vertices is $x$, i.e., $|J|=x$.

You shouldn't confuse sets and their cardinalities, i.e, the number of elements. These are entities of different type. The cardinality of $A$ is often denoted by $|A|$. For example, the phrase "A∪B is A+B, which is n" in post #1 should in fact say, "$|A\cup B|$ is $|A|+|B|$, which is $n$".

c4nn3t said:
So I'd start out saying for a graph Z, each vertex J has degree greater than x/2. This must mean that K is also > x/2. I want to show that K is <= 2 between any two vertices
You cannot say, "$K$ between two vertices" because $K$ is the set of all edges. Picking two concrete vertices does not change $K$; it is defined once for all in the scope of this problem.

c4nn3t said:
So once I prove that the graph is diameter <=2, how does that lead to it being connected?
Hmm, if I can walk from any vertex to any other vertex in at most two steps, why does it follow that I can walk from any vertex to any other vertex in any number of steps?

c4nn3t said:
Ultimately, I state that the G is connected if J has connectivity n-1. Do I need to consider if the graph is Eulerian?
No.

Pick two vertices $u$ and $v$. Let $U$ be the set of vertices adjacent to $u$, and let $V$ be the set of vertices adjacent to $v$. Then by assumption $|U|>x/2$ and $|V|>x/2$ where $x=|J|$. Obviously, $U\subseteq J$ and $V\subseteq J$. Do you think you can apply Part A from post #1 to this situation?
 
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