For posts seeking explanation or clarification of a specific step in a proof. "Please explain this proof" is off topic (too broad, missing context). Instead, the question must identify precisely which step in the proof requires explanation, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.
Questions tagged [proof-explanation]
12360 questions
115
votes
7 answers
If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable
I am going over a tutorial in my real analysis course. There is
an proof in which I don't understand some parts of it.
The proof relates to the following proposition:
($S$ - infinite $\sigma$-algebra on $X$) $\implies $ $S$ is…
Belgi
- 23,614
104
votes
9 answers
Division by $0$ and its restrictions
Consider the following expression:
$$\frac{1}{2} \div \frac{4}{x}$$
Over here, one would state the restriction as $x \neq 0 $, as that would result in division by $0$.
But if we rearrange the expression, then:
$$\begin{align}
\frac12\div\frac4x &=…
Devansh Sharma
- 1,145
66
votes
7 answers
Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples]
The following question was asked on a high school test, where the students were given a few minutes per question, at most:
Given that,
$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$
and,
$$Q(x)=x^4+x^3+x^2+x+1$$
what is the remainder of $P(x)$…
joeblack
- 1,023
53
votes
4 answers
What is the explanation for this visual proof of the sum of squares?
Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
Nitin
- 2,988
51
votes
12 answers
Does Monty Hall logic apply to this real world situation?
I recently posted a tweet claiming I had encountered a real life Monty Hall dilemma. Based on the resulting discussion, I'm not sure I have.
The Scenario
I have 3 tacos (A,B,C) where tacos A and C are filled with beans, and taco B is filled with…
Will Cole
- 615
48
votes
4 answers
A subgroup of a cyclic group is cyclic - Understanding Proof
I'm having some trouble understanding the proof of the following theorem
A subgroup of a cyclic group is cyclic
I will list each step of the proof in my textbook and indicate the places that I'm confused and hopefully somewhere out there can clear…
Amateur Math Guy
- 1,481
47
votes
5 answers
In a proof by contradiction, how do we know the assumption is the cause of the contradiction?
In a proof by contradiction, how do we know the assumption is the cause of the contradiction? And not just the result of some other property more fundamental to numbers?
In other words, how can we be sure we arrived at the contradiction because…
Stephen
- 3,796
37
votes
10 answers
Why does this way of solving inequalities work?
Here is what I had to prove.
Question: For positive reals $a$ and $b$ prove that $a^2+b^2 \geq 2ab$.
Here is how my teacher did it:
First assume that it is in fact, true that $a^2+b^2 \geq 2ab$. Therefore $a^2+b^2-2ab \geq 0$ . We have $(a-b)^2$…
The Cryptic Cat
- 1,457
35
votes
8 answers
Reasoning that $ \sin2x=2 \sin x \cos x$
In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$.
What's the math behind this formula that made it true? Can someone explain?
Commander Shepard
- 1,872
35
votes
12 answers
Is this a valid proof that there are infinitely many natural numbers?
I remember reading a simple proof that natural numbers are infinite which goes like the following:
Let $ℕ$ be the set of natural numbers.
Assume that $ℕ$ is finite. Now consider an arbitrary number $K$, where $K$
is the largest number in…
groov
- 439
33
votes
3 answers
Can't find the flaw in the reasoning for this proof by induction?
I was looking over this problem and I'm not sure what's wrong with this proof by induction.
Here is the question:
Find the flaw in this induction proof.
Claim $3n=0$ for all $n\ge 0$.
Base for $n=0$, $3n=3(0)=0$
Assume Induction Hypothesis: $3k…
user262291
- 1,469
31
votes
6 answers
Right adjoints preserve limits
In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an adjunction, then if $D:I\to \mathcal{D}$ is a diagram that…
Bruno Stonek
- 12,973
29
votes
4 answers
A function with a non-zero derivative, with an inverse function that has no derivative.
While studying calculus, I encountered the following statement:
"Given a function $f(x)$ with $f'(x_0)\neq 0$, such that $f$ has an inverse in some neighborhood of $x_0$, and such that $f$ is continuous on said neighborhood, then $f^{-1}$ has a…
Ran Kiri
- 980
29
votes
12 answers
How to prove that $\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$ without squaring both sides
I have been asked to prove:
$$\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$$
Which I can easily do by converting the LHS to index form, then squaring it and simplifying it down to get 2, which is equal to the RHS squared, hence proved.
However I know…
adam
- 557
25
votes
3 answers
Proof that the continuous image of a compact set is compact
Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(X)$ is a compact set.
I know that this question may be a duplicate, but the problem is that I have to prove this using real analysis…
Lessa121
- 1,358
- 2
- 10
- 23