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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3answers
39 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
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1answer
63 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
59 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
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1answer
71 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
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4answers
71 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
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2answers
63 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?
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1answer
359 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
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1answer
83 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
51 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
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1answer
64 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
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1answer
47 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
votes
0answers
50 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
116 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
97 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
23 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
433 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
votes
1answer
36 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 ...
2
votes
2answers
44 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 ...
12
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5answers
239 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
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1answer
64 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
21 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
65 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
votes
0answers
30 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
49 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
votes
1answer
34 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
54 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
219 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
3answers
161 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
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1answer
61 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 ...
0
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0answers
54 views

Verify my proof: if $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive.

Could someone verify my proof and my writing? Proposition: If $R$ on $X$ is transitive then the weak and strict preferences I and P derived from R are also transitive. Definition 1: A binary ...
0
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1answer
12 views

question about vacuous truth and function

I'm confusing about vacuous truth. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ where $f(n)=2n$. we can calculate function values if $n$ belongs to domain. but what if it does not? The value of ...
1
vote
7answers
77 views

Induction proof of $1 + 6 + 11 +\cdots + (5n-4)=n(5n-3)/2$

I need help getting started with this proof. Prove using mathematical induction. $$ 1 + 6 + 11 + \cdots + (5n-4)=n(5n-3)/2 $$ $$ n=1,2,3,... $$ I know for my basis step I need to set $n=1$ but I ...
0
votes
2answers
61 views

If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$

Suppose that $k$, $n$, and $d$ are integers and $d$ is not $0$. Prove: If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$. You may not use the theorem stating the following: Let ...
1
vote
1answer
63 views

Can someone verify my proof by contraposition?

This is a problem from Discrete Mathematics and its Applications Is there a way to tell right away what type of proof to use or does that just come with practice (build intuition - oh here i ...
2
votes
0answers
25 views

Sorting for maximum mean squared successive difference

I have a set of numbers and I have to order them for maximum MSSD (mean squared successive difference). For example, if I have the ordered set {1,2,3,4,5,6} this would give me an MSSD of ...
4
votes
2answers
158 views

If $X$ is finite and $R$ is a complete and reflexive binary relation on $X$, then $M(R, S) \neq \emptyset$ on any $S \subset X$ iff $R$ is acyclic.

Could you help me to verify my proof and my writing? Definition 1: A binary relation $R$ on $X$ is complete if, for all $x, y \in X$ such that $x \neq y$,$xRy$ or $yRx$ or both and reflexive if, for ...
2
votes
3answers
151 views

Real Analysis Proofs

I am taking a Real Analysis class using the textbook Analysis with an Introduction to Proofs, $5^{th}$ Ed. by Steven Lay. So far I am not understanding the proofs at all. Does anyone know of any good ...
2
votes
2answers
110 views

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction [duplicate]

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction My proof so far: Let $P(n)$ be $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ Base Case $P(1):$ LHS = $1^3 = 1$ ...
3
votes
0answers
81 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
0
votes
1answer
79 views

Prove that $nCr = n(n-1)(n-2)\cdots(n-r+1)/ 1\cdot2\cdot3 \cdots r$ is an integer for all positive integral $n$ and for all integers $r \geq 0$.

Prove that $nCr =\frac{ n(n-1)(n-2)\cdots(n-r+1)}{ 1\cdot2\cdot3 \cdots r}$, is an integer for all positive integral values of $n$ and for all integers $r \geq 0$. Can someone please explain it to ...
-2
votes
2answers
217 views

Inequality $\prod\limits_{r=1}^{\infty}(1+(\frac{1}{2})^r)<\frac 52$ [closed]

Prove this inequality. $\prod\limits_{r=1}^{\infty}\left(1+\left(\frac{1}{2}\right)^r\right)<\dfrac 52$ I have tried to prove it using induction but it is not coming.
2
votes
1answer
184 views

How to remember all the proofs in mathematics

I have a problem where I forget the proof of a theorem after some time without reworking it out. However, my teacher said that he was able to prove a theorem even without reworking it out for a long ...
0
votes
3answers
232 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
1
vote
2answers
36 views

My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$

Is it reasonable to prove the following (trivial) theorem? If yes, is there a better way to do it? Let $x, y \in \mathbb{R}$. Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$. ...
0
votes
2answers
105 views

Attempt to proof the Cantor-Bernstein theorem

I've found a proof of the Cantor-Bernstein theorem in Kleene's 'Introduction to Metamathematics' (1952) in §4 Thm A. I must admit I don't understand its essence but I was wondering if the proof could ...
1
vote
1answer
31 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
1
vote
1answer
55 views

How to display one to one correspondence?

This is a problem from Discrete Mathematics and its Applications Here is the book's definition of countable/not countable For 2a, I came up with the fact that the set is countably infinite. What ...
3
votes
2answers
75 views

Prove that $ A \subseteq B \iff \mathcal{P}(A) \subseteq \mathcal{P}(B) $.

I'm going through Velleman's How To Prove It and I'm currently on section 3.4 which deals with techniques for proofs involving conjunctions and biconditionals. The title of this question is from one ...