For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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1answer
39 views

If $2^{k} + 1$ is prime, prove that $k$ has no other prime divisors than $2$. [duplicate]

I am trying to prove this by contradiction: Assume $2^{k} + 1$ is prime. By definition of odd number $2^{k} + 1$ is odd because $2^{k} + 1 = 2\cdot 2^{k-1} + 1$ Then $2^{k} + 1 \pmod{2} \equiv 1 ...
1
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1answer
24 views

Prove that if $p > 3$ is a prime then $2(p-3)! \pmod{p} =-1 \pmod{p}$.

Prove that if $p > 3$ is a prime then $2(p-3)! \pmod{p} =-1 \pmod{p}$.. I am totally lost; at first I thought this could be done by induction, but unfortunately this is not possible (at least I ...
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0answers
19 views

Proving with a given definition that if $|A|=|B|$ then $A,B$ are equivalent (with induction but without using the induction hypothesis)

Let $A,B$ be finite sets, we'll say the sets are equivalent if $|A\setminus B|=|B\setminus A|$. Prove with the above definition that if $|A|=|B|$ then $A,B$ are equivalent. Suppose ...
0
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1answer
17 views

Proof of floor function identity.

Let $f(x) = \lfloor x \rfloor$ and let $l$ be the greatest integer $\le x$ How do I prove $l + 1 > x$ I see that: $x \ge \lfloor x \rfloor = l$ No complete answers, just hints
2
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1answer
26 views

Proof of existence of “peaks” in The Sequential Compactness Theorem

According to Bolzano–Weierstrass theorem: How do we guarantee that EVERY sequence of real numbers has either infinitely or finitely many "peaks"? (Note that a subsequent is ordered in a way of ...
2
votes
3answers
48 views

Prove that $0 < 1$. Prove that $ab = 0 \implies a = 0$ or $b = 0$.

Proof: There exists $a = 0$ (For every $b$, an element of the set of positive numbers, such that: $b > a$) $$a + b > 0 \implies b > 0 \implies a < b.$$ Thus, we have shown that $0 < ...
1
vote
1answer
45 views

Is the solution to this elementary number theory problem correct?

Problem: A natural number $n$ is called nice if the following properties hold: • The expression is made ​​up of 4 decimal digits; • the first and third digits of $n$ are equal; • the second and ...
0
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3answers
50 views

Proof by Contrapositive?

i am having trouble proving the statement below using Proof by Contrapositive. I have negated the statements as required and then i prove that $n$ is odd if and only if $7n+4$ is odd. However, from ...
2
votes
3answers
26 views

Prove that $\sqrt x$ is Lipschitz on $[1, \infty)$

Prove that $\sqrt x$ is Lipschitz on $[1, \infty)$ I want to show that $|f(x) - f(y)| \leq L |x - y|$ So $|\sqrt x - \sqrt y| = \frac{|x - y|}{\sqrt x + \sqrt y} \leq \frac{1}{2}|x - y|$. I can ...
1
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0answers
13 views

Cavalieri's principle and integrals

Let $f$ be a continuous function an $[a,b]$. Let $P\subset R^2$ be the figure under the graph of $f$ and $D \subset R^3$ a solid figure obtained by rotating a plane curve around $x$-axis. Using ...
0
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1answer
27 views

Rectangles in one dimension

I have to prove the following proposition : Show that the intesection of two rectangles in $\mathbb{R}^{n}$ is either the vaccum or is another rectangle. My attempt: I one is embeded in the other ...
1
vote
2answers
42 views

Proving if $|A|\ge 4 \vee |A|\le 2$ then $|A+A|\neq 4$ with direct, contradiction and contraposition

Prove if $|A|\ge 4 \vee |A|\le 2$ then $|A+A|\neq 4$. $A$ is some set and we define $A+B=\{a+b|a\in A, b\in B\}$, $A$ is some subset of the reals. In a direct proof and proof by contradiction I'd ...
5
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2answers
57 views

Proving that $\sin1 $(radian) is irrational without using Taylor Series Expansion.

In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational, and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but ...
3
votes
1answer
38 views

Verifying a Vector Space Via Given Axioms

Let $X$ be the collection of all sequences $\{\alpha_n\}_{n=1}^{\infty}$ of scalars from $\mathbb{K}$ such that $\alpha_n=0$ for all but a finite number of values of $n$. Define addition and scalar ...
0
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1answer
30 views

Using set theory to prove a function problem

I begin with: $$A = \{a \le x < x_0 | f(x) = 0 \}$$ $$B = \{x_0 < x \le b | f(x) = 0 \}$$ Let $c = \sup A$ and let $d = \sup B$ First to prove $f(x) > 0$ for $x \in (c, d)$ I will ...
1
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2answers
52 views

How to get to $5^3 \geq n^3$ in the proof by contradiction?

This is the same problem asked here. - Next step to take to reach the contradiction? Here is it again. I understand the solution - how you want to get to the fact 100 divides n^2 and then go ...
0
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2answers
36 views

Prove that $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$

Prove: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ Proof: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ ...
1
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3answers
33 views

Proving by contradiction that if $a\in\mathbb Q,b\in \mathbb R\setminus \mathbb Q$ then $a+b\in \mathbb R \setminus \mathbb Q$

I'm trying to prove by contradiction that if $a\in\mathbb Q,b\in \mathbb R\setminus \mathbb Q$ then $a+b\in \mathbb R \setminus \mathbb Q$, I already proved it with contra position and a direct proof ...
0
votes
1answer
43 views

Elementary Operations on Sets

Let $X$ be a set with subsets $A$ and $B$. Prove: a). $X \setminus (X \setminus A) =A$. $X \setminus A$ is the set of all points of $X$ which do not belong to $A$. Given $p \in X$, we will show that ...
0
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1answer
39 views

Antiderivative of an even function

I'm faced with an issue in terms of antiderivatives of even and odd functions. Define $f \in C[-a,a]$ where $a>0$. Let $f$ be an even function on $[-a,a]$. We wish to show that $$\int_{-a}^a ...
3
votes
1answer
55 views

Proof that $\sum_{i=1}^n{1} = n$ for all $n \in \Bbb Z^+$

It seems obvious that $$\forall n \in \Bbb Z^+, \sum_{i=1}^n{1} = n $$ However, I'm having trouble coming up with a formal proof for this. Given a concrete number like $4$, we can say that ...
1
vote
4answers
62 views

Show that if $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$.

Show that if $m$ and $n$ are integers such that $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$. I am not sure where to begin.
0
votes
2answers
46 views

prove that 2 does not go into $n^2 – 2$ without a remainder for odd $n$.

Prove that $2$ does not go into $n^2 – 2$ without a remainder for odd $n$. How do I approach this?
0
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1answer
24 views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
0
votes
1answer
26 views

Proving the existence of a Bijection between Cartesian Products of Sets by Induction

Prove by induction that for any sets $A_1, \ldots , A_n$, there is a bijection from $(((A_1 \times A_2) \times A_3) \times \ldots \times A_n)$ to $A_1 \times (A_2 \times ( \ldots (A_{n-1} \times A_n) ...
1
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2answers
43 views

Proving rigorously that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ using divisibility definition

Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ After a bit of algebra I get that $$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$ So now how ...
0
votes
1answer
62 views

What is the contraposive of this statement?

I have to prove the negation of this statement: $$\forall a,b,c\in\mathbb{Z}{\;if\;a\;|\;b\} $$ But the fact that there is a "and" is very disturbing. I think that I am missing something because my ...
0
votes
1answer
36 views

how to prove $pr_i(\alpha \setminus \beta) \supseteq pr_i\alpha \setminus pr_i\beta$

For those who are not familiar with the syntax $pr_i \alpha = \{ pr_i(a,b) / a \alpha b \} \text{ for }\alpha \subseteq A \times B$ which is same as $\begin{cases} (x= pr_1 \alpha) \Leftrightarrow ...
4
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0answers
36 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
0
votes
4answers
64 views

Prove that a continuous real function with finite limits is bounded

$f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Assume that $\lim_{x \rightarrow \pm \infty} f(x)$ exist and are finite. Prove that $f$ is bounded. So to show that $f$ is bounded, I ...
2
votes
1answer
42 views

How to adapt proof by contradiction showing that a sqrt(2) is irrational for sqrt(20)?

This example is from Discrete Math and its Applications I understand the steps the author is taking. First he assumes sqrt(2) is rational meaning that there exists integers a, and b such that ...
1
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0answers
17 views

Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications For the phrase "a and b have no common factors" , does that actually mean a and b have no common factors other than 1? I feel like this would ...
13
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3answers
345 views

How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...
0
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1answer
26 views

$\lim_{x\to x_0 ;x\in X} f ( x)$ exists if f is a uniform continuous function and $x_0$ is an adherent point

Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists. Proof Take any ...
0
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0answers
46 views

Proving that an infinite sum of irrationals is irrational

First of all, I know this question may be closed because it is off topic, but I do have a valid question. Problem: Is is possible to prove that an infinite sum of distinct and different irrational ...
-2
votes
0answers
32 views

Proof: A+B is upper triangular [closed]

Assume A and B are nxn matrices. Prove that A+B is upper triangular.
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2answers
38 views

Is $\sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous?

$f(x) = \sin (e^{x^2} + \cos(3x^{2} + 5))$ on $[0, 1]$ uniformly continuous because: Proof: $f(x)$ is a continuous function on $[0, 1]$, which is a closed interval, so $f$ is uniformly continuous on ...
4
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0answers
24 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
1
vote
1answer
39 views

Clarification on Cantor Diagonalization argument?

My book is Discrete Mathematics and its Applications. This is its section on Cantor's Diagonalization argument I understand the beginning of the method. The author is using a proof by ...
0
votes
1answer
20 views

How to prove the $i$th coordinate of $u$ in the basis equals $(u,e_i)$?

Let $e_1, e_2, e_3$ be an orthonormal basis. How to prove that for any vector $u$, $$u = (u, e_1)e_1 + (u, e_2)e_2 + (u, e_3)e_3,$$ i.e., the $i$th coordinate of $u$ in the basis equals $(u,e_i)$?
2
votes
3answers
58 views

How to approach this proof problem, what proof to use, what assumption to use?

This is a problem from Discrete Mathematics and its Applications Here is the definition of rational that my book uses Usually when I approach this type of a problem, I can find a type of proof to ...
0
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0answers
24 views

Would it be necessary to have another proof within the proof by cases in this problem?

This is a problem from Discrete Mathematics and its Applications I am using Proof by Cases. This is my book's definition on it. Here is my work so far I tried to leverage without of generality ...
1
vote
1answer
39 views

Induction on the size of the set?

Show that every non-empty finite set of real numbers has a maximum. (Hint: induction on the size of set). I'm not exactly sure how to approach this. I'm familiar with induction, but I don't know ...
0
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1answer
25 views

Proof of Z-transform of n

How can I prove the following Z-transform: $$ Z\{n\} = \frac {z} {(z-1)^2} $$ As a tip, I was told to use the 'Multiplication in time'-property of the Z transform, which is the following: $$ ...
0
votes
3answers
36 views

Next step to take in this proof by contradiction?

This is a problem from Discrete Mathematics and its Applications Here is my work so far It's similar to this other question I had Next step to take to reach the contradiction?. I am assuming ...
2
votes
3answers
63 views

Can someone verify my direct proof that if $A$ is a subset of $B$, $A\cup B = B$?

This is a problem from Discrete Mathematics and its Applications I am trying to use a direct proof to do this problem. Here is my book's explanation/section on direct proof Here is my work so ...
0
votes
0answers
11 views

Subsets being cones

I am trying to self-study convex optimization and still trying to get into the gist of it. There is a question in my text as follows: Let $V$ be the set of sequences whose terms are contained in ...
1
vote
4answers
140 views

Proving that $p_1p_2\mid n$ iff $p_1\mid n$ and $ p_2\mid n.$

Let $p_1$, $p_2$ be distinct primes. Using the Fundamental Theorem of Arithmetic prove that a natural number $n$ is divisible by $p_1p_2$ if and only if $n$ is divisible by $p_1$ and $n$ is divisible ...
1
vote
1answer
43 views

Question about maps in partially ordered sets

I am struggling with a proof related to the mapping of posets. Let $P_1$and $P_2$ are posets, and $f$ an order preserving map from $P_1$ to $P_2$. $$g(Z):= f^{-1}[Z]$$ $g$ goes from the set of all ...