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]
11824 questions
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How to prove that the curve $x^2+y^2-3=0$ has no rational points?
There is a question 6.20 from the Book of Proof by Hammock that reads as follows: We say that a point $P=(x,y) \in R^2$ is rational if both x and y are rational. More precisely, P is rational if $P=(x,y) \in Q^2$. An equation $F(x,y)=0$ is said to…
mont2223
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Can it be proved by contrapositive?: If $x | y^2$ then $x | y$
We want to show that if $x|y^2$ then $x|y$. Is this contrapositive proof correct?
Proof by contrapositive: Assume $x \not| \;y$ and $x,y \in Z$. Thus, $y=xk+r$ for $k \in Z, r \in N, r < k$ by definition of not divides (there is a remainder).…
mont2223
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Could someone clarify this picture in Spivak, illustrating $|x - a| < \frac{|a|}{2}$?
It's not really clear to me what the picture is trying to illustrate. If we mark $a$ to be the largest vertical bar, then how many of these unit bars is $a$ supposed to be? $4$?
polite proofs
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To estimate $e$ more accurately, how would you divine to define $e = 2^c$?
I understand all the algebra, but not "let $e = 2^c$". This feels like the key step. I've been staring at this the whole day, and I would've never been able to prognosticate this substitution.
James Stewart, Calculus 7th ed. 2011. Not the Early…
user851668
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Quick question about showing a function is one to one
Let $f : ℕ → P(ℕ)$ be given by $f(n) = {k*n | k ∈ ℕ}$. (P(ℕ) is the power set of the set A.)
Is f injective?
The answer to this question goes like this:
$f(n) = (n,2n,3n,4n,5n...)$
$min({ n,2n,3n,...})$ is $n$, as $n<2n<3n<....$
Suppose…
mathstudent288
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How can I prove that $m
I'm having trouble taking this test, I apologize if the question is duplicated and I appreciate any indication of content. My math base is not that good, so please be clear. If $0 < m < n$, can I prove:
$$m < n \Rightarrow m^2 < n^2.$$
William Teixeira
- 113
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Prove that the following statements are logically equivalent:
$s_1: A \backslash B = \emptyset$ and $B \backslash A= \emptyset$
$s_2: A \cup B = A \cap B$.
I'm not sure if what I have is a correct way to show this proof, but I have the following so far:
Suppose $A \backslash B = \emptyset$ and $B \backslash…
Not2Scary
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How to prove using direct, contradiction, and contrapositive
If $m$ and $n$ are odd, then $m+n$ is even.
For direct proof I said $m=2a+1$ and $n=2b+1$ so $m+n=(2a+1)(2b+1) =2a+2b+2 =2(a+b+1)$ which is even I think.
For contradiction, I think it begins as "If $m$ and $n$ are odd, then $m+n$ is odd." but I…
Belle
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Is Felix Klein approach the same as Arnold on Abel's Theorem?
Arnold has a geometric approach to prove Abel's theorem, this is explained in V. B. Alekseev's book Abel's Theorem in Problems and Solutions: Based on the Lectures of Professor V.I. Arnold . It's quite impressive.
I just found that Felix Klein also…
athos
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Prove that if x,y,z ∈ R, then |x − y| ≤ |x − z| + |z − y|.
Prove that if x,y,z ∈ R, then |x − y| ≤ |x − z| + |z − y|.
I know that in order to get the inequality I start off with |x-y| = |(x-z)+(z-y)| and my final result should be |x-y| ≤ |(x-z)+(z-y)| using the triangle inequality in the last step. Can…
user824294
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Proof - what is the name of this proposition?
Can anyone tell me the name of this proposition, and give a proof it it is straightforward?
$$ \prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j $$
tam63
- 197
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Prove that if x,y > 0 then (1/2)(x+y) ≥ √(xy). For which x,y does equality hold?
Answer:
x+y ≥ 2√(xy) ⇔ (x+y)^2 ≥ 4xy ⇔ x^2+y^2+2xy ≥ 4xy ⇔ (x−y)^2 ≥ 0, which is true. Equality holds when x=y.
I've solved it until (x−y)^2 ≥ 0, but I don't understand "which is true".
If y is larger than x won't it not be true? e.g. 14 - 16 = -2.…
user30200
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About finite decimal
It is well known that $0.0\dot9=0.1$ and we say $0.1$ is finite decimal. And $0.0\dot9$ is a repeating decimal.
This is concluded that $\text{finite decimal } = \text{ infinite decimal}$ which is a contradiction.
How is this possible?
TCLee
- 37
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Elaboration on theorem about pairwise distances of points
In a book I own, it states:
Let $m_{ij}, i,j=0,1,\dots ,n,$ be nonnegative real numbers with $m_{ij}=m_{ji}$ for all $i,j$ and $m_{ii} = 0$ for all $i$. Then points $\mathbf{p}_0,\mathbf{p}_1,\dots ,\mathbf{p}_n \in \mathbb{R}^n$ with $||…
donguri
- 320
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Is the statement "$p$ implies $q$" logically equivalent to the statement "$p$ implies only $q$"?
I am confused if the statement "$p$ implies $q$" logically equivalent to the statement "$p$ implies only $q$"?
Assuming that the two said statement is logically equivalent, then the truth value of the statement ...
"If $a^2=b$ and $b>0$, then…
AYA
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