Questions tagged [proof]

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3 votes
2 answers
197 views

Linear dependence of two vectors is a necessary and sufficient condition for an equality

This is an example in the textbook. To prove: for any real number** $a_1, a_2, \cdots, a_n, b_1, b_2, \cdots b_n$, linear dependence of the two vectors $\left(a_1, a_2, \cdots, a_n\right)$ and $\left(...
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6 votes
3 answers
1k views

The integer ababab (a,b>0) is always divisible by 7, without remainder

The integer $ababab$ $(a>0,b>0)$ is always divisible by $7$, without rest. I tried to prove this by: ...
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-1 votes
1 answer
68 views

Why can't Mathematica confirm this simple identity? [closed]

It is well known that for positive integer $n\geq2$, $$ \sum_{k=1}^{n-1} \frac{(-1)^{k-1}(k-1)!^2}{(n^2-1^2)\ldots(n^2-k^2)}=\frac{1}{n^2}-\frac{2(-1)^{n-1}}{n^2 \binom{2n}{n}}. $$ This identity ...
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2 votes
4 answers
533 views

Maximum sum of product of two arrays [closed]

Any math proof for maximum sum of product of two array achieved by maximum times maximum A = {5, 2, 1, 3}; B = {1, 2, 3, 4}; ...
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1 vote
1 answer
29 views

Why does Mathematica not return "True" for Correlation function equality expression?

I wish to check the invariance property of correlation. As described in this wiki section Wiki - Mathematical Properties of Correlation. A quote from the Wiki page I wish to test is below. That is, ...
-2 votes
1 answer
134 views

Proving an integral identity [closed]

Assuming $f$ is continus on $[0,\pi]$, show that $$\int_{0}^{\pi }xf(\sin x)dx=\frac{\pi }{2}\int_{0}^{\pi }f(\sin x)dx$$ I tried to demonstrate it using integration by parts, but I did not succeed. ...
0 votes
0 answers
51 views

What is the proper way to feed this function as input and test if my statement is true?

$f(S)$ = {$n∈N$: $n=m²$ for $m∈S$} $S$ = $1,2,3$ Square each element in S Now, $S= 1, 4, 9$ All elements in $S$ are counting from $1$ to $N$. In this case $N=3$ $f(S)$ = {$n∈N$: $n=m²$ for $m∈S$} ...
0 votes
1 answer
97 views

Prove Geometry automatically [closed]

If given a geometric problem, can Mathematica solve it automatically? Is there any software that can solve geometry problems automatically?
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4 votes
1 answer
74 views

Trigonometric identity not simplifying

I have a pair of numbers {x,y} (Reals or Integers) for which I want to confirm symbolically that ArcTan[y/x]+ArcTan[x/y]==Pi/2 Here is my attempt: ...
  • 1,549
1 vote
1 answer
72 views

Can Mathematica check if I correctly applied an algebraic transformation?

Mathematica has tons of functions to perform algebraic manipulations. But is there a way to use it to check if I didn't make mistakes in my own algebraic manipulations. As a real world example: [...
7 votes
1 answer
206 views

Using logical equivalence to prove PL statements

I'm trying to use those Logical equivalence as axioms to prove some PL statements, In this case I followed the examples in the documentation that didn't use the build-in logic functions$\{\text{And}[,]...
  • 225
6 votes
1 answer
131 views

FindEquationalProof to prove divisor theorem

I'm trying to implement proofs of concepts for Equational Proofs on some basic number theory theorems. One such example is: "let a and b be positive integers and let d = gcd (a, b). If t divides ...
2 votes
0 answers
135 views

How to prove subspaces of function spaces?

In teaching Linear Algebra, I would like to prove or disprove whether a nonempty subset of a vector space, is a subspace. For this, I would show that the subset is closed under both addition and ...
2 votes
0 answers
117 views

FindEquationalProof with implications

Motivated by this question in mathematics, my Mathematica question is essentially this one, but alas the answer to that isn't quite adequate. I'm not interested in verifying that a proof is valid or ...
7 votes
1 answer
374 views

FindEquationalProof with Logic in Wolfram Mathematica

I try to "make" proof in Wolfram Mathematica. Thats a proof: if a->b and b->c then a->c I tried ...
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4 votes
1 answer
97 views

What does this graph mean?

Mathematica 11.3.0 includes a new command FindEquationalProof having good prospects. Studying it, I consider a somewhat modified example from the help ...
  • 20.4k
1 vote
1 answer
46 views

Proofs and exponents to the N power

I'm still a beginner at Mathematica and I'd like to understand how to use it for proofs. Here is a simple one on exponent properties (a*b)^n == a^n*b^n I tried ...
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6 votes
1 answer
231 views

Implementing the field axioms with FindEquationalProof?

I am fiddling around with FindEquationalProof, currently trying to prove some basic statements for fields. I have a set of axioms which almost constitutes the field theory axioms: ...
9 votes
1 answer
193 views

Proving uniqueness of group identity element

Start, as in the Mathematica 11.3 documentation, with: ...
  • 11.6k
2 votes
1 answer
114 views

When Reduce yields False for a system of inequalities, can I take this as a formal proof? [closed]

As part of the proof of a proposition in an economics paper, I need to show that a system of inequalities in four variables is inconsistent. Because the system is complicated, I cannot show this ...
1 vote
3 answers
88 views

Can you determine values in a set of integers from a known set sum and set product

Question: There is a set of numbers defined as {A, B, C, D, E} The sum of the set is 49 (A+B+C+D+E = 49) The product of the set is 13000 (ABCDE = 13000) Can you determine the values of A, B, ...
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4 votes
1 answer
963 views

Proof of the Dirac-$\gamma$ matrices identity

Given the matrices $\gamma_{k}=\begin{bmatrix} O & -i\sigma_{k}\\ +i\sigma_{k}& O \end{bmatrix}$ where $\sigma_{k}$ is the $k^{th}$ Pauli matrix $\gamma_{4}=\begin{bmatrix} I^{2} &0 \...
  • 670
2 votes
2 answers
791 views

How to "Prove" this summation result?

I have this messy function with $n$, $k$, $i$ integers: $$ r(\rm n,k)=\frac{k 2^{1-2 \rm{n}} (2 k)! (-2 k+2 \rm{n}+1) (2 \rm{n}-2 k)!}{(k!)^2 \left(1-4 (i-k)^2\right) ((\rm{n}-k)!)^2} $$ I want to ...
  • 7,083
4 votes
0 answers
121 views

How to use Resolve to prove ForAll function

I want to prove that ∀n ∈ N,Mod[9^n - 8 n - 1, 64] == 0 I think I should use Resolve to prove it ...
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6 votes
1 answer
831 views

How to extract steps from FullSimplify? [duplicate]

The answer proposed to the original question doesn't solve it, I need to understand the results of TraceInternal when used on FullSimplify as a sequence of steps, which has not been resolved for me. ...
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10 votes
2 answers
483 views

Symbolically prove that two expressions are identical

I encountered this problem when trying to reproduce the result of this paper. (The relevant parts are all included in the preview i.e. the 1st page of the article. This link is just given as ...
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5 votes
2 answers
221 views

Proving that Reverse[Reverse[x]]]=x

It is clear that for any list $x$, Reverse[Reverse[x]]=x. I want to have Mathematica tell me that this is true. I have tried entering ...
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0 votes
2 answers
95 views

How to prove that an equation can be divided by any n?

I'm new to Mathematica and came across with the problem. I can't prove that (for example) 2*n is always even no matter what n is....
5 votes
0 answers
199 views

Theorem Proving

It's a hard for me to write english so I'll try to go straight to my problem. There is abstract predicate f, that takes two parameters. There are also two other ...
6 votes
1 answer
225 views

Fundamental question about capabilities of Mathematica to represent abstract mathematics [closed]

I have an fundamental question about what Mathematica can and cannot do. I have a book which presents a certain physical theory in an axiomatic manner. The axioms make heavy use of mathematics. Some ...
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17 votes
3 answers
1k views

Proving (or at least 'being told by Mathematica') that Sqrt[2] is irrational?

I realize that Mathematica is not specifically an automated theorem prover. However, this article: http://www.wolfram.com/products/mathematica/newin6/content/EquationalTheoremProving/ Suggests that ...
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4 votes
2 answers
329 views

Using Mathematica to confirm Bernoulli's inequality

I have several challenges that I want to confirm is true. I have chosen this one because it is rather simple (proof by induction). There are times when I do not want to spend ages trying find proofs. ...
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2 votes
2 answers
149 views

Performing an inductive proof

I have an expression which depends on a variable n and am certain it evaluates to 1 for all n with n=1/2,1,3/2,... I want to do an inductive proof that this is true but the actual formula is too ...
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5 votes
0 answers
264 views

Can I define an axiomatic (Boolean algebra) system and prove theorems using Mathematica?

The general question is Can I define an axiomatic system and prove theorems using Mathematica? The more concrete one is about Boolean algebra. I consider this axiomatic Boolean algebra system (wiki)....
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33 votes
1 answer
1k views

Proving the hairy ball theorem using xAct

I would like to formally prove the hairy ball theorem in Mathematica, initially just for $S^2$, and then see about generalizing. An approach I thought about to use the xAct package to define $S^2$ ...
1 vote
1 answer
3k views

Is this function convex?

How can I determine convexity of the function f = Log[ x, 1 + (x^a - 1) (x^b - 1)/(x - 1)] with the parameters $a,\,b$ belonging to the interval $(0,1)$ in ...
  • 20.4k
-2 votes
3 answers
417 views

Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer

Find integer values of p such that $(2^p - (2^2)(3^2))/ (3^3)$ is an integer.
7 votes
1 answer
3k views

Solving functional equations in Mathematica

I am trying to solve functional equations in Mathematica, but got nowhere. Just a simple example: $$f(x+y)^2=f(x)^2+f(y)^2$$ for all $x,y \in R$, assuming f is a real-valued function. How should I ...
3 votes
2 answers
345 views

How can I prove an equation has a certain number of real roots?

I have this code ...
5 votes
3 answers
379 views

Proving a recurrence in Mathematica

I have $$j_n=\int_0^1 x^{2n} \sin(\pi x)dx.$$ How do I show that $$j_{n+1}= \frac{1}{\pi^2}(\pi- (2n+1)(2n+2)j_n)\, ?$$ I keep getting a recurring integration by parts and I can't simplify it. ...
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8 votes
4 answers
3k views

How to prove an expression by method of mathematical induction?

I want to prove the expression $$1^2 + 2^2 + 3^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$$ by method of mathematical induction with Mathematica, but I do not know how to start. How do I tell ...
  • 4,413
13 votes
0 answers
537 views

Proving (self) similarity with Mathematica - Reccurrence Plots, Similarity Plots etc

I posted this question in math.se but given the sheer tumultuous number of questions that keep appearing on math.se and also given that I am trying to accomplish this in Mathematica, I thought I'd ...
  • 5,271
12 votes
3 answers
2k views

Using FindInstance to Prove No Solutions Exist

For a small amount of background, I am currently working on an undergraduate research project in Combinatorial Geometry and I'm working on a case analysis for embedding spherical simplicial 2-...
15 votes
1 answer
2k views

Proving inequalities with Mathematica

Question summary: I would like to learn some tips and tricks on how to prove inequalities with Mathematica. I'm studying various inequalities in triangle that have the form $R+ar + bs\geq 0$, where $...
  • 4,876
12 votes
7 answers
3k views

Trying to prove that $x\sin(\frac{\pi}{x})\ge\pi \cos(\frac{\pi}{x})$ for $x\ge 1$

As an intermediate step, consider the function: f[x_] := x Sin[Pi/x] I want to prove that this function is increasing for $x\ge 1$. This can be done with the ...
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