Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

Questions with this tag are about the presentation of a mathematical proof. Questions might include:

  • Should I include [x-mathematical detail] at [y-part of this proof]?
  • Is the following a sufficient proof of [x-mathematical tidbit]?
  • I have written the following proof, could I somehow improve it, does it have good flow/can I improve readability?

But this tag is not for asking someone else to write a proof for you, or for how to answer some question. Questions such as: My professor asked me to prove the Pythagorean theorem and I don't know how to begin are not to have this tag.

This tag is intended for use along with other, more "mathematical" tags. A question about the writing of a proof in abstract algebra, for example, should have as well. This tag can be used along with the proof verification tag.

See here for a useful set of guidelines for writing a solution.

15776 questions
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How to prove the triangle inequality axiom for the metric system $d(x,y) = \lvert\,x - y\,\rvert$

In other words, How does one prove $d(x,z) \leq d(x,y) + d(y,z)$ for $d(x,y) = \lvert\,x - y\,\rvert$ given $x, y, z \in \mathbb{R}?$
Robot0110
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Simple intuitive example as to why this proof technique is wrong

Suppose we want to prove the following: If f(x) has property A then f(x) also has property B I was told that if you prove if f(x) has property B then f(x) also has property A, this does not prove the original statement. Are there simple examples…
jsguy
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Does an assumption need to be true after a proof?

If I start a proof by saying: Assume $a^3=b$ and end up proving something with it, will the proof hold for when $a^3\neq b$? Would that be a valid proof for all $a$ and $b$ baring any other domain issues?
GenuinePlaceholder
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Inequality Proof by Contradiction

I need help setting up the following proof: Prove that $0 ≤ a < b$ implies $0 ≤ a^2 < b^2$ and $0 ≤ a^{(1/2)} < b^{(1/2)}$. I am thinking proof by contradiction is the right method, I just dont know how to start it.
Bridget
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Prove that Every number divides 0 while the only number that 0 divides is itself.

this problem from An Inquiry-Based Introduction to Proofs v1. by Jim Hefferon I has problem in the second part that only 0 can divide 0 I think it is interminate form and when put 0/0 in wolfram it shows it's undefined Is book wrong or I miss…
Lingnoi401
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If an equality is true for any finite N then we can take the limit

Suppose we have the following equality. $\mu (F_{N})= \sum_{i=0}^{N}G_{n}$ for any finite $N$. Is it true that we can then the limit if this relation then? i.e lim $\mu (F_{N})=\sum_{i=0}^{\infty}G_{i}$
user415535
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On structuring the presentation of a proof with respect to logic

I have done the following exercise. Show that if $x_{1}$ and $x_{2}$ are linearly independent and $z>0$ then $zx_{1}$ and $zx_{2}$ are linearly independent. Now I am not sure how to regard this in general. One way would be; Let $x_{1}$, $x_{2}$ be…
user415535
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Sigma notation using induction

Suppose that n∈N, $$\sum_{k=1}^n (2k+1) = n^2+2n$$ Base Case:n=1 ⟹2∗1+1=3=12+2∗1 the base case holds true I.H, Assume its true for $$\sum_{k=1}^{n} (2k+1) = n^2+2n$$ Then; $$\implies\sum_{k=1}^{n+1} (2k+1) = n^2+2n$$ $$\implies\sum_{k=1}^{n+1}…
TheGamer
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Prove: if n is odd, then kn is odd

Stuck on this proof...any help is appreciated. Here is what I have thus far. Suppose n is odd, then n = 2j + 1 for some integers j, k, and m. then m(2j + 1) = 2jm + m. Since the product of two integers is an integer, let k= jm. => 2k + m I'm sure…
electr0hed
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Write a formal proof of the statement

Write a formal proof of the statement "for all rational numbers $b, c$ if the equation $x^2 + bx + c = 0$ has a rational solution $r$, then any other solution $s$ of this equation is a rational number". We can use the two following predicates: Let…
HKT
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Proof that $y\cdot x^y=x\cdot y^x, x\ne y$ has no solutions in $x,y\in\mathbb{R} ;x,y>0$ has no solutions

$y\cdot x^y=x\cdot y^x$; $x\ne y$; $x,y\in\mathbb{R} $; $x,y>0$ has no solutions $$y\cdot x^y=x\cdot y^x$$ $$\frac{x^y}{x}=\frac{y^x}{y}$$ $$x^{y-1}=y^{x-1}$$ $$\ln(x^{y-1})=\ln(y^{x-1})$$ $$(y-1)\ln x=(x-1)\ln y$$ $$\frac{\ln x}{x-1}=\frac{\ln…
Jacob Claassen
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How to say that something is a function?

I have the following writing problem: Let's say that $f,g\colon A \rightarrow B$ are given functions and I want to proof that $h = (f,g) \colon A \times A \rightarrow B \times B$ is a function. Formally, I have to take $(a,b),(a,c)\in h$ and show…
HeMan
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I need a simple example of proving a TONCAS.

I know the definition of the acronym is "The Obvious and Necessary Condition Are Sufficient". I am not exactly clear as to what it means to prove a theorem's TONCAS. Does it simply mean prove the converse since the necessary condition must be also…
TheLast Cipher
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Formal proof of this simple equation?

What is the formal proof of $1^{-1}=1$. Intuitively it makes a lot of sense but how would the formal proof go ?
Andres Romero
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Given p and q and (p ∧ q ⇒ r), use the Fitch system to prove r.

I am having trouble beginning to learn how to create proofs. I can understand the individual rules well enough, but putting them together seems to create a whole which is greater than the sum of its parts. I am not understanding why an Assumption of…
master__mind
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