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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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Writing Proof Examples

What are examples of magic mountain an circle diagram proofs for 2nd - 5th grade math problems? I want to teach how to prove word problem outcomes of addition and subtraction.
McLean
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Let $x \in \mathbb{Z}$. Prove that $5x-11$ is even IFF $x$ is odd.

Let $x \in \mathbb{Z}$. Prove that $5x-11$ is even IFF $x$ is odd. I know that for an IFF proof we prove it directly, and then prove it again by the contrapositive. I don't know how to prove this directly, so in a normal direct proof I would use the…
Wng427
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Simple Proof - ∃n∈ Z: ∀k ∈ Z: n < k

How to prove these 2 (one is true, the other one is false): ∃n∈ Z: ∀k ∈ Z: n < k ∀n∈ Z: ∃k ∈ Z: k < n where Z = {0,±1,±2,...} EDIT1: What I have so far: . ∀n ∈ Z: ∃k ∈ Z: k < n Pose n = x et k = x - 1, For any x ∈ Z, there will always be a term…
nnnn
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What are the two ways to prove existence?

I know proof by example is one way, but I cannot remember what the other way is. I thought maybe it was proof by contradiction, but I cannot think of an example for the life of me.. so I'm not satisfied with assuming.
fractionofyoursmile
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Difference between biconditional iff and equality

I'm currently learning about mathematical proofs. I got confused about the concept of biconditional iff and equality "=". Suppose I want to prove c(a+b)=ca+cb. Is it correct to prove it by showing that c(a+b)$ \Leftrightarrow $ca+cb where I'll show…
Blake
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How to prove we must insert parentheses(brackets) with powers

Question: When we want to add three numbers, say $a + b + c$, we don’t bother inserting parentheses because $(a + b) + c = a + (b + c)$. But with powers, this is not true - ${(a^b)}^c$ need not be equal to $a^{(b^c)}$ - so we must be careful. Show…
user30200
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In order for a proposition to be completely stated, how much detail needs to go into describing it?

During my study of mathematics so far, I had come to realize that how a proposition is stated can be just as important as the proof itself. For example, when I was working on various propositions on the relationships of limits, one I had worked on…
GovEcon
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Proof of if A, then B or C

I am wondering how to prove if A, then B or C. I saw one proof is to show if A is true and B is false, then C is true. I think it’s true but can’t reason why. Could someone show me the logic behind it? Also wonder are there other ways to prove it,…
Coco
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What is the logical statement for C ⊆ A ∪ B in an equation?

Set C is a subset of A union B. 1 is in set A, then also a union with B gives {1, 2, 3, 4, 5, 6} We know – A U B is { x | x є A) v (x є B) C ⊑ is a subset of A U B which means every element in A U B is also C. Ɐ є { x | x є A) v (x є B) Is this…
Nichole Reed
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Proposed method to prove conjectures in continuous settings

I thought of a way to prove conjectures in continuous settings, assuming they are true at the two ends points, To demonstrate it, I will show how it works with the power rule from diffrential calculus.$$$$ Just a basic recap, the power rule says…
Ofek Tevet
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Correct choice of proof for showing that there is an integer satisfying $5^n=n^5$

I am wanting to start a proof for the following statement: Prove that there is an integer $n$ such that $5^n=n^5$. Would proof by construction be an appropriate method to use? Then would proof by induction be an appropriate method to use?
user430574
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Why do mathematicians approach axiomatic proofs like this?

When doing proofs, I keep a tab open on 'Advice for students for learning proofs', this guidelines helps me take the right first steps when looking at statements. But, with axiomatic proofs, I am on a shaky foundation. See this proof of 2.1.2 (a)…
Kevin Dudeja
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Find three sets $A$, $B$, $C$, each of them non-empty, such that $(A\cap B)\cup C=A\cap(B\cup C)$ and $(A\cap B)\cup C\neq A\cap(B\cup C)$

I need to find three sets for both statements I have above. I have tried drawing Venn diagrams and shading appropriately then adding numbers in each shaded region to try and guess and check for sets $A$, $B$ and $C$ but am unsure how to find an…
Bruh
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Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous at a. Prove $f \circ f: \mathbb{R} \to \mathbb{R}$ is continuous at $f(a)$.

I am having a bit of difficulty with this proof. I know $|x-a| < \delta \implies |f(x) - f(a)| < \epsilon$. I know $f$ is only continuous at a, so do I need to prove f is continuous at $f(a)$ before moving on with the proof? If yes, I'm not sure…
Ann Watson
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Inverse implies surjection and follow-your-nose proofs

(I'm posting this question with my own answer, to show a nice calculational proof for one of the examples in Luke Palmer's blog post Follow Your Nose Proofs.) In what follows, $A$ and $B$ are sets, with $f,g : A \rightarrow B$ and $h : B \rightarrow…
MarnixKlooster ReinstateMonica
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