# 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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### Under which conditions discretization of convex\concave function is submodular?

Say, I have $f(x)$ with $x \in [0,1]$, then by discretization I mean $f(x_h)$ with $x_h \in \{0, h, 2h, \dots, 1\}$. I know about Lovasz extension, but it works in other way: given submodular ...
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### Proving that $\lceil f(x) \rceil$ $=$ $\lceil f(\lceil x \rceil )\rceil$ when $f(x) =$ integer $\implies x =$ integer

On P. 71 in 'Concrete Mathematics' the following Theorem is given: Let $f$ be any continuous, monotonically increasing function on an interval of the real numbers, with the property that \begin{...
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### If a set $A$ is uncountable , and a set $B$ is countable then $A \times B$ is uncountable.

I prove it by contradiction. Let $A \times B$ is countable. It means we can list down the all the ordered pairs of $A \times B$. So if ordered pairs of the form $(a,b)$ are countable (where $a \in A$ ...
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### Recurrence equation for sequence of vectors

Consider recurrent formula for a sequence of numbers $(y_n)$ (either real or complex): $$a_k y_{n+k}+a_{k-1}y_{n+k-1}+\cdots+a_0y_n=\sum_{i=0}^k a_i y_{n + i} = 0$$ It's known that the exact explicit ...
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### How many distinct ways are there to $2$-color the $8$ vertices of a cube?

How many distinct ways are there to $2$-color the $8$ vertices of a cube, with colorings only considered distinct up to rotation?
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### Applications of tensor product of graphs (modelling of Internet Graphs)

I was going through the book Handbook of Product Graphs, by Richard Hammack, Wilfried Imrich, Sandi Klavžar. Somewhere in book, they mentioned the following lines : One of the applications of tensor ...
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### In how many ways can 10 different things be distributed to 4 persons if 2 are to receive 2 things and the others are to receive 3 things?

I have no idea how to answer this question, I did a lot of research on trying to figure it out but every answer is so different. I would prefer something along the lines of using combinations and ...
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### Conditional probability in independence and mutually exclusive events.

This thread shows that if two events are to be mutually exclusive and independent, one of them should have zero probability. I worked the following example that seems to contradict conditional ...
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### Find $x$ in $1!+2!+\ldots+100!\equiv x \pmod{19}$

Here I come from one more (probably again failed) exam. We never did congruence with factorials; there were 3 of 6 problems we never worked on in class and they don't appear anywhere in scripts or ...
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### Function over non-numerical sets

Considering a finite lexicographically ordered set, for example, $\{a, b, c, d\}$ called $A$ with $A$ as domain and codomain of a function which returns the element with right shift of 1 over A, how ...
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### Finding the generating function of a recurrence relation in dependence of a variable

Given this inhomogeneous linear recurrence relation of 2nd order : $F_n = F_{n-2} + a$ for $n \geq 2$ with $F_1 = 1$ and $F_0 = 0$ How do I find the generating function of this recurrence ...
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### How to investigate the relationship between range and payload?

I am interested in learning about the relationship between range and payload for an electric aircraft. How do I use math to investigate the relationship between range and payload for an electric ...
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### If $n,r\in \Bbb{Z}^+$ and $2^{r-1}+2-r \leq n < 2^r+1-r$, find $r$ in terms of $n$ in closed form.

For integer $r$ and $n$, consider the relations $$2^{r-1}+2-r \leq n < 2^r+1-r$$ To eliminate possible pathological cases for small $n$, take both $n$ atnd $r$ to be at least $3$. I'd like to ...
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### Is a relation between A and B the same as a mapping from elements of A to subsets of B?

The way I always saw it was that a relation is a subset of $A \times B$, or a collection of ordered pairs $(a,b)$, where $a \in A$ and $b \in B$. Is there any meaningful distinction between the two ...
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### Finding the cardinality of sets [closed]

Find the cardinality of each of the following sets. a) {x, {x}} b) {a, {a}, {a,{a}}} c) P({a, {a, {a}}}) d) P({∅})
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### Show $\sum_{i=1}^{n+1} 2^ii = 2^{n+2}n+2$ , for all integers $n\ge 0$ using induction.

How would i solve this problem?, i tried and went through the entire process is this correct and how would i factor the induction part. $\sum_{i=1}^{n+1} i\cdot 2^i = n\cdot 2^{n+2}+2$ , for all ...
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### What is the complement of a product of two sets?

I am given this information: Suppose $A=\{1,2,3\}$, $B=\{3,5\}$, $C=\{1,2,4,6,9\}$ and $U = \{0, 1, 2, 3, 4, 5, 6,7,8,9\}$. Enter "T" for each true, and "F" for each false statements. There ...
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### Sum of products expansion of basic Boolean function: $F(x,y) = \bar{y}$

So I have a question about this very basic-looking sum of products expansion. My professor has this particular example in his lecture slides but I can't quite wrap my head around this. I don't ...
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