# 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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### Why is the unique readability of wff's important?

I am reading Classical Mathematical Logic by Epstein. The author defines: $L(\neg, \rightarrow, \vee, \wedge, p_0, p_1, ...)$ i. For each $i=0, 1, 2, ..., (p_i)$ is an atomic wff, to which we ...
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### PCA for antisymmetric matrix

PCA (Principle Component Analysis) is often used to convert a symmetric matrix to lower dimension one. My question is whether there is semiliar method for antisymmetric matrix? As we known, the ...
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How does one interpret this ZFC Union axiom? I can't quite understand what is meant after "There exists some elements y for all elements z"? I'm also wondering if the x is a typo. $\exists y \... 2answers 40 views ### How to formally prove whether this function is onto or not?$ K(x) = x^ 2$where$x \ge 0$.$K(x) = x^2.$The domain and range of this function comprise of non-negative real numbers. If it were real numbers instead of "non-negative" real numbers, then it seems easy to prove it by ... 1answer 7 views ### Spanning 2-regular subgraphs in even regular graphs. Theorem: Every regular graph of positive even degree has a spanning 2-regular subgraph. This was taken from Corollary 5.10 of ETH Zurich's notes on graph theory. The proof constructs a Eulerian tour,... 4answers 58 views ### Probability that the second throw of a fair die exceeds the first A player throws an ordinary die and records the score$A$. The player then throws the die again and again records the score,$B$. if$B>A$then we set a score for this player. What is the ... 3answers 3k views ### Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math) Antisymmetric:$\forall x\forall y[ ((x,y)\in R\land (y, x) \in R) \to x= y]$reflexive:$\forall x[x∈A\to (x, x)\in R]$What really is the difference between the two? Wouldn't all antisymmetric ... 1answer 1k views ### Discrete Laplace transform Yesterday ago I was reading how the Laplace Transform can be interpreted as the continuous analog of the discrete functional dependance of the power series $$f(x) = \sum a(n) x^n$$ This is to say, $$L\... 5answers 7k views ### What is the difference between discrete and continuous mathematics? I am studying computer science and this has me absolutely flummoxed. The definition I can find is that discrete data is countable and that continuous is uncountable. Examples are given stating that ... 1answer 33 views ### Prove that in a tree #leaves + #nodes of degree 2 \geq \frac{n}{2} I am trying to solve the following problem: Let T = (V,E) be a Tree with n = |V| nodes. Let b denote the number of leaves and z the number of nodes with degree 2. I want to show that$$ ... 0answers 12 views ### How many vertices will the following graph have? 35 edges Degree at least 3 Thought n(3+) = 70 5n=70 n=14 For an integer result. What if degrees range from 3 to something? 0answers 22 views ### Choosing values in a strong induction The sequence s0,s1,s2... is defined by s0=1 and for all integers n>0,$s(n)=s(⌊n/2⌋)+s(⌊2n/5⌋) + n.$Prove, using strong induction, that S(n) > 4n for all integers n>=3. To my knowledge, I only have ... 1answer 76 views ### Prime Factorization Question involving Product of Consecutive Terms I came across this question while doing some research at an REU this summer. It was supposed to be just a small part of a larger proof, but we've been stumped on it for a while. I don't have much of a ... 1answer 382 views ### Suppose that A is a countable set. Show that the set B is also countable if there is an onto function f from A to B. Is my logic correct/accepted? Let A be a countable set. Let f:A->B, surjective.$\exists$g:A->N, bijective. Using definition from 1.$\exists$h:N->B, surjective.$\therefore$B is countable by ... 6answers 115 views ### Prove by induction$3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$My question is: Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever$n$is a nonnegative integer. I'm stuck at the basis step. If I ... 4answers 161 views ### Disprove that “if$p$is a prime number, then$2^p-1$is also a prime number”? We can see manually that$2^p-1$is not prime. As$2047$is not a prime.$2^{11} = 2048$. But I'm unable to figure out a formal way of disproving the statement. 1answer 32 views ### Need help understanding a proof (Bipartite Graph) I was reading lecture notes of graphs(from MIT 6042) and am having trouble understanding this proof: I can't understand ... 1answer 27 views ### venn diagram category help. For each list of categories, draw a venn diagram that shows the relationship among the categories. One of the categories describe the universal set, and the others describe the various loops inside ... 1answer 905 views ### Disjunctive normal form (BOTH dnf and cnf) example help T or ( not x and y ) or ( x and y ) In class we went over examples of how many things can be both DNF and CNF... eg. (not z) or y can be thought of as both (not z) or y .... dnf ((not z) or y)... 0answers 30 views ### Prooving reflexivity and antisymmetry [on hold] Good day. I have to prove reflexivity and antisymmetry for integers a and b. I understand that reflexivity is a≤a and antisymmetry is if a≤b and b≤a then a=b. However, aside from stating simple ... 1answer 24 views ### Counting the number of Eulerian trails in a connected, directed graph I can't find anything about this online, and I'm beginning to suspect it's a hard problem. I know that counting the number of circuits is #P-complete, but I don't need the number of circuits; I need ... 0answers 22 views ### Disjoint subset problem Show there exists two disjoint subset A, B of$\{1, 2, 3, \cdots , 10\} = [10]$such that $$s(A) = \sum_{i\in A}i = \sum_{i\in B}i=s(B)$$ namely, the sum of elements in A equals that of B. Really not ... 1answer 22 views ### Distribute 20 million$ among 4 companies with some constraints

20 million is to be invested in 4 companies A, B, C, D. The minimum amount for investments are 1, 2, 3, 4 million respectively. How many different investment strategies are available if An ...
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### In how many ways we can arrange 12 people in a row if 5 men are constrained to sit next to each other together?

In how many ways we can arrange $12$ people in a row if $5$ are men and they must sit next to each other? My approach I consider $5$ men as one entity and so now there are $8$ people to be seated ...
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### Prove that ∀n≥1, (1/(1⋅3))+(1/(3⋅5))+(1/(5⋅7))+…+(1/(2n−1)(2n+1)) =( n/(2n+1))| [on hold]

Prove that ∀n≥1, (1/(1⋅3))+(1/(3⋅5))+(1/(5⋅7))+...+(1/(2n−1)(2n+1)) =( n/(2n+1))| So, I understand that the proof must display that (1/(2n−1)(2n+1) is equivalent to (1/(2n−1)(2n+1). Would I solve ...
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### Spanning trees of the complete graph minus two edges

Here is the following problem: What two edges should one remove from the complete graph $K_n$ so that the number of the spanning trees of the new graph is as small as possible? One can solve this ...
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### Bijective/Injective function mapping

I have a quick question about bijective functions. Let's say I want a function that maps $$f:(0,1] \rightarrow [0,\infty)$$ I can say $$f(x) = 1- \frac1x$$ right? And if I want an injective ...
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### Algorithm for multiplying infinite decimals?

What is the (best) algorithm for multiplying two real numbers based on their decimal expansions? Obviously the algorithm can't be completed but I mean an algorithm that will successively approximate ...
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### Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
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### How to draw prefix and postfix binary tree?

I have drawn these two binary trees. The ordered set of numbers is [-9, -5, 0, 1, 5, 7, 8, 10, 11] The first one is in prefix order and the second is in postfix order, but I'm not sure if my ...
I have a function in the form of $V = a[1-exp(t-t_0)]$ and $V_0=0$. I'm using this formula in discrete system and I need to discretize this formula and solve it every T seconds and get the $V_{k+1}$ ...