# Tagged Questions

The study of discrete mathematical structures. Consider using a more specific tag instead, such as: (combinatorics), (graph-theory), (computer-science), (probability), (elementary-set-theory), (induction), (recurrence-relations), etc.

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### All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
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### Help with combinatorial proof of binomial identity: $\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$

Consider the following identity: $$\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$$ Consider the set $S$ of size $2n-2$. We partition $S$ into two sets $A$ ...
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### Proving Set Operations

I'm trying to prove that if $A$ is a subset of $B$ then $A \cup B = B$, but I am having trouble trying to proves this mathematically. I know that since $A$ is a subset, then $A$ has an element $x$ ...
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### Strange Recurrence: What is it asymptotic to?

So I have the following recurrence relation for the growth rate of an algorithm: $T(n)$ = time taken by algorithm to solve problem of size n: $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$ Clearly this ...
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### Generating Functions- Closed form of a sequence

We are given the following generating function : $$G(x)=\frac{x}{1+x+x^2}$$ The question is to provide a closed formula for the sequence it determines. I have no idea where to start. The denominator ...
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### Prove with induction that $11$ divides $10^{2n}-1$ for all natural numbers.

$$10^{2(k+1)}-1 = 10^{2k+2}-1=10^{2k}\cdot10^{2}-1$$ I feel like there's something in that last part that should make it work, but I can't grasp it. Am I missing something obvious? Am I going in the ...
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### Need a counter example for cycle in a graph

Could anyone give a counter example for that theorem : A graph G has exactly one vertex of degree $1$, then it contains a cycle. I am so confused. I wonder that may I give a counter example ...
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### Is there a formula for finding the number of nonisomorphic simple graphs that have n nodes?

As my subject line asks, is there a formula for finding the number of nonisomorphic simple graphs there are with n nodes, outside of trial, error, and enumeration over max degrees of vertices? Thanks ...
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### Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$.

For $a,b \in \mathbb{R}$ define $a \sim b$ if $a - b \in \mathbb{Z}$ I don't understand how I'm suppose to prove this: Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$ Also can you ...
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### Definition of a countable set

What is the proper definition of a Countable Set?
### Equivalence relation $(a,b) R (c,d) \Leftrightarrow a + d = b + c$
Suppose $A$ is the set composed of all ordered pairs of positive integers. Let $R$ be the relation defined on $A$ where $(a,b) R (c,d)$ means that $a + d = b + c$. (a) Prove that $R$ is an ...