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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.

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Proving with well ordering principle

I'm having a very rough time wrapping my head around with well-ordering principle, and how to use it with proofs.. For example: If S is a subset of Z(integers) which is bounded below, then there is a natural number k so that $$ S + k \subset N…
dendritic
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Prove that the following pairs of sets have equal cardinality:

(b) $\mathbb Z$ and the set $\{x \in \mathbb R \mid (\exists n \in\mathbb Z)(x = 2^n)\}$ (c) $\{0, 1\} \times \mathbb{N}$ and $\mathbb{Z}$ (d) $\{0, 1\} \times \mathbb{N}$ and $\mathbb{N}$ For each of these questions, my logic is as follows; We…
Hearth
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If $ax + by = 8$, what is $\operatorname{gcd}(a, b)$?

Our instructor has given us this problem: If $ax + by = 8$, what is $\operatorname{gcd}(a, b)$? I'm confused. Is it not just $8$? Since, say, $\operatorname{gcd}(a, b) = n$, so there must exist some $x, y$ such that $n= ax + by$. Do I need to…
Rdewolfe
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Proving that a Binary Tree of $n$ nodes has a height of at least $\log(n)$.

For a homework assignment, I need to prove that a Binary Tree of $n$ nodes has a height of at least $log(k)$. I started out by testing some trees that were filled at every layer, and checking $log(n)$ against their height: when $n = 3$ and $h = 1$,…
Carcigenicate
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Proving an Iff Statement

Suppose we had a function defined over the complex numbers: $ f(x)= \begin{cases} 1&\text{if } x\in\mathbb{R}\\ 0&\text{if } x\not\in\mathbb{R} \end{cases} $ And we are asked to prove that $f(x)=1$ iff $x$ is real. Obviously, if $x$ is real, then…
Arcturus
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Prove that $f^{-1}\left(U_1\times\cdots\times U_\kappa\right)=\bigcap_{i=1}^\kappa \left(f_i\right)^{-1}\left(U_i\right)$

Im working through Bloch's Proofs and Fundamentals and exercise 4.3.11 is Let $B$ be a set, let $A_i,\cdots,A_\kappa$ be sets for some $k\in\mathbb{N}$ be a subset for all $i\in\left\{1,\cdots,\kappa\right\}$ and let $f:B\rightarrow …
bjd2385
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For every $A\in \mathcal {P}(U)$ there is a unique $B\in \mathcal{P}(U)$ such that for every $C\in \mathcal{P}(U)$, $C\cap A=C-B$

Pls help me out with the proof: For every $A\in \mathcal {P}(U)$ there is a unique $B\in \mathcal {P}(U)$ such that for every $C\in \mathcal {P}(U), C \cap A=C-B$. For the existence part, we have to figure out some $B$ which works for all $C$. What…
user233467
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Proof of Transitivity

Let R be the following relation of x and y on Z where 3x + y is even. I can seem to get to the form of $3x + z$ when I am doing algebraic manipulations if this equation. I have $3x + y = 2k$ and $3y + z = 2i$ for some $k, i \in Z$. I substituted…
Justin
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Suppose $A$ is a subset of $B$ and $B$ is a subset of $C$ and $A$ is equinumerous with $C$. Prove $B$ is equinumerous with $C$.

Definition I use: $A \sim B$ means $A$ is equinumerous with $B$ which means there is a $f\colon A \rightarrow B$ that is a bijection. My goal is to prove the following, Suppose $A \subseteq B \subseteq C$ and $A \sim C$. Prove that $B \sim C$. I…
ChemDude
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Question about choosing cases in a proof by cases

Prove that for every integer $n \ge 8$, there exist nonnegative integers $a$ and $b$ such that $n = 3a + 5b$. Proof: Let $n \in \mathbb Z$ with $n \ge 8$. Then $n = 3q$ where $q \ge 3, n = 3q + 1$ where $q \ge 3$ or $n = 3q + 2$ where $q ≥ 2$. We…
keys
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Power set equinumerosity. Is this proof correct?

So I'm trying to prove the following, Prove that if $A\sim B$ then $\mathscr{P}(A) \sim \mathscr{P}(B)$. Here's how I started out to prove there is a function that is injective: Suppose $A \sim B$. Then we can choose a function $f:A\rightarrow B$…
ChemDude
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Using existential instantiation on a universally quantified given

I'm trying to prove the following exercise of How to Prove it: A structured Approach (Section 3.4, exercise 19): Suppose A, B and C are sets. Prove that A $\triangle$ B and C are disjoint iff A $\cap$ B = B $\cap$ C. The answer to the proof goes…
jviotti
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Examples on how to give a proof or a counterexample of a statement

Examples; Prove or give a counterexample of the following statements,with quantifiers: 1) For each non-negative number s, there exists a non-negative number t such that s≥t 2) For each non-negative number t, there exists a non-negative number s such…
Jianluca
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Do I have the right start for this proof?

I'm trying to prove the following, Suppose R is a partial order on $A$, $B\subseteq A$, and $b\in B$. Prove that if $b$ is the smallest element of $B$, then it is also the greatest lower bound of $B$. My givens and goals so far are: Givens $b$…
ChemDude
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What would the correct English description be for the difference for 1/3

If you measure a task & it takes 3 seconds, then the next time you do the same task, it takes you 1 second, is the difference 200% or 67%? Or would you say the difference is 200% because 3-1=2 or 200% better -- but the percentage of difference is…
Peter
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