## Is this a valid proof of Hall’s Theorem on System of Distinct Representatives?

Hall’s Theorem states that for sets $$A_1,…,A_n$$ if and only for all $$J\subseteq \{1,…,n\}$$ we find $$|\bigcup_{j\in J}A_j|\geq|J|,$$ then we can choose $$x_i\in A_i$$ for all $$i$$ such that $$\{x_1,…,x_n\}$$ is a system of distinct representatives for $$A_1,…,A_n.$$ To me it seems easiest to some inductive argument (or a contradiction that is similar to induction), however, in my graph theory class we’re working with Flow Algorithm’s, so the goal is to solve it using the Max-Flow-Min-Cut Theorem. I’ve seen one proof using that theorem, but I think I’ve came up with another, which to me seems more natural. I was wondering if my proof is correct. Here’s the proof.

$$\textbf{Proof:}$$

Throughout $$x$$ shall refer to an element of some $$A_i.$$ We set up a capicated network as follows $$\overrightarrow{E}=\{(s,A_i):\forall A_i\}\cup\{(A_i,x):\forall A_i\text{ and }x\in A_i\}\cup\{(x,t):\forall x\in \cup_{i=1}^n A_i\}$$ with capacities for $$(s,A_i)$$ and $$(x,t)$$ all $$1$$ and all other capacities infinite. This is ‘depicted’ in the following link, where any unwritten capacity is infinite.

View post on imgur.com

Now we suppose that we have a minimal network cut $$U.$$ So the sum of capacities of edges leaving $$U$$ is minimal. If $$A_i\in U,$$ and $$x\not\in U$$ for sum $$x\in A_i,$$ then $$(A_i,x)$$ is an edge leaving $$U$$ with infinite capacity, and $$\{s\}$$ has smaller capacity than $$U,$$ impossible. So $$x\in U$$ for all $$x\in A_i$$. We compute $$\text{capacity}(U)=\sum_{A_i\not\in U}\text{capacity}(s,A_i)+\sum_{x\in U}\text{capacity}(x,t)=|\{A_i\}\backslash U|+|\{x:x\in U\}|$$ $$\text{capacity}(U)\geq|\{A_i\}\backslash U|+|\{x\in A_i: A_i\in U\}|\geq|\{A_i\}\backslash U|+|\{A_i\in U\}|=n.$$ Thus since $$\text{capacity}\{s\}=n$$ we find that $$\{s\}$$ is a min cut, and we can find a system of distinct representatives. $$\blacksquare$$

## ping at University halls of residence

One computer has a ssh server running on Linux, and I can login on the localhost with : ssh myuser@localhost.

The Linux computer IP is : 10.188.116.102

The other computer has Windows 7.

The Windows computer IP is : 10.188.40.210.

Now, I’m trying to connect to the Linux computer from the Windows computer, both on the desk, but ping has negative answer :

either

ping -t 10.188.40.210

from the Linux computer, or

ping -t 10.188.116.102

from the Windows computer answer :

Destination Host Unreachable

Also, here is the result of ifconfig on the Linux computer :

enp0s25: flags=4163<UP,BROADCAST,RUNNING,MULTICAST>  mtu 1500         inet 10.188.116.102  netmask 255.255.0.0  broadcast 10.188.255.255         inet6 fe80::6ef0:49ff:fe2a:8dc9  prefixlen 64  scopeid 0x20<link>         ether 6c:f0:49:2a:8d:c9  txqueuelen 1000  (Ethernet)         RX packets 171603  bytes 242463049 (231.2 MiB)         RX errors 0  dropped 0  overruns 0  frame 0         TX packets 63232  bytes 6620049 (6.3 MiB)         TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0         device interrupt 16  memory 0xfc200000-fc220000

and the some network details on the Windows computer :

IP   : 10.188.40.210 mask : 255.255.0.0 DHCP : 10.188.0.1

## What Are The Facilities Would Be At Banquet Halls?

Normally banquet halls provide a spacious hall, sitting arrangements based on the client's requirements and food. For example, Naveen hotel Hubli has well spacious banquet halls and has outdoor Banquet Hall designed only for special occasions. Be it a serious meeting or large video conference, every facility they provide to run your work smoothly in a lavishly well-appointed space. Lakeside Garden Banquet is of 500-seater and makes an ideal venue for Reception and Wedding parties. And…

What Are The Facilities Would Be At Banquet Halls?

## How To Pick Banquet Halls In Hubli?

Finding the perfect venue is the toughest part of getting married, business meetings and conferences. We should check first when we are searching for banquets was Ambience, house catering and the banquet hall is suitable for our expectations. For example, Naveen hotel in Hubli offers best Banquets in Hubli. They designed some halls for Meeting and conferences. Whether it is a small function or large one, we ensure there is every facility to run your work smoothly in a lavishly well-appointed…

How To Pick Banquet Halls In Hubli?