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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19 views

Practice Examples of Proofs by Induction, Direct/Indirect Method

I'm learning about proofs in school, quite a few different sorts (but not geometry ones), but the teacher is teaching by slides mainly, not books. The main ones are proof by ...
4
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1answer
47 views

Help getting started on a proof

I honestly have no idea where to start with the following proof, and I was wondering if anyone could help me get started. I don't want the whole idea, I just need to know where to start with this ...
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1answer
30 views

Finding a minimum weight spanning tree? [duplicate]

Letting W be the weighted graph created by taking a complete graph K5 on five vertices 1, 2, 3, 4, 5 with the weight of each edge {x,y} given by ({x,y})=x+y, How would I find a minimum weight ...
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1answer
36 views

Verify my proof: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $

Could someone verify my proof and my proof-writing? Proposition: for two positive natural numbers $x$,$y$ , if $ x + y = 2$ , then $ x = y = 1 $ Proof: Suppose $ y $ is any positive natural ...
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0answers
99 views

Can $3x^3+3x+7$ be cube number? [duplicate]

Can $3x^3+3x+7$ be cube number when $x \in \mathbb{N}$? My conjecture is that the answer is no, but I don't know how to prove it. Can anybody help me to solve this problem?
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1answer
19 views

Proof with Cartesian coordinates.

Let $S_b := \{(x,y) \in\mathbb R^2 | y = 3x + b\}$ where $b\in\mathbb R$. Give a direct proof that if $(r,s)\in\mathbb R^2$, then there exists a $b\in\mathbb R$ such that $(r,s) \in S_b$. I have ...
3
votes
1answer
66 views

Uniform convergence on an interval

Let $a<c<b$. Let {$f_n$} be a sequence of functions converging uniformly on $[a,c]$ and $[c,b]$. Prove that {$f_n$} converges uniformly on $[a,b]$ My attempt: Intuitively, I see that {$f_n$} ...
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3answers
39 views

Proof that the greatest common divisor of (a, a+2) is 2 if a is even and 1 if a is odd

Some help would be great on this, my teacher hasn't explained how to construct proofs to us, he just keeps doing them for us in class. I have at the beginning: Let a be even. Since the sum of two ...
2
votes
2answers
238 views

Prove that if a and b are positive integers, then there exists integers x and y such that 1/lcm(a,b)=x/a+y/b

My professor has not taught us the technique of writing proofs, he just continues to do them for us in class. So I am really stumped on this proof. Any help is greatly appreciated!
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3answers
50 views

How would I show that Tn=3^n + 2 is a solution to the recurrence?

Would anyone be able to help me or give me some advice on the following problem: Consider the recurrence with $T_0 = 3$ and $T_{n+1} = 3T_n - 4$ for all $n \in \mathbb{N}$. How would I show that ...
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0answers
16 views

Using a direct proof to prove circumscribed shapes.

I am looking at this problem: Use a Direct proof to show that if A is a circle circumscribed by a square B, and the square B is circumscribed by a Circle C, then the area of Circle C is twice the ...
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3answers
47 views

The only positive divisor of both $a$ and $a + 1 $ is $1$

Prove that if $a \in \mathbb Z$ then the only positive divisor of both $a$ and $a + 1$ is $1$. When I saw this statement I didn't understand it. The only way that I can see it being true is if a is a ...
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0answers
39 views
+50

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$ Prove for $1: e_{ii}A=Ae_{ii}$ and for $2: e_{ij}A=Ae_{ij}$ where $i\le j$ Now for 1, I understand ...
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3answers
61 views

Prove by induction that… $1+3+5+7+…+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$

I'm not too sure exactly how to approach this question. Would anyone be able to give me any helpful advice or some sort of direction? I have a little problem with induction. Prove by induction that: ...
4
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3answers
695 views

Is there a better alternative to the phrase, 'it holds that'?

The following phrases abound in my writing: There exists [whatever] such that [whatever]. For all [whatever] it holds that [whatever]. Lately, I've been feeling that the phrase 'it holds that' is ...
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0answers
31 views

Questions on logic behind “proof by contradiction”

I'm trying to understand the logic behind "proof by contradiction" and hoping that I can clear up a few things in this post. First of all, suppose I have a proposition $P$ and from this I can imply ...
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1answer
33 views

How can I show that a and b are odd in this contradiction proof?

Statement: suppose a,b belongs to Z (integers). If 4/(a^2+b^2) then a and b are not both odd. By proof of contradiction I assume that a and b are both odd. If a^2 and b^2 is odd then by definition a ...
0
votes
2answers
39 views

Is this contradiction proof correct?

Statement : suppose $a,b$ belongs to $\mathbb{Z}$ (integers). If $4/ (a^2 + b^2)$ then $a$ and $b$ are not both odd. Proof by contradiction: Assume that if $4/(a^2 + b^2)$ then a and b are both odd. ...
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3answers
50 views

Simplifying a proof by contradiction: if $a\equiv 1\bmod 5$, then $a^2\equiv 1\bmod5$

Prove the following either by Direct Proof or by Contraposition: Suppose $a\in\mathbb{Z}$, if $a\equiv 1\pmod 5$, then $a^2\equiv 1\pmod5$ Suppose $a\equiv 1\pmod 5$ Then $5|\left(a-1\right)$, ...
2
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1answer
44 views

Proving a Problem involving Fibonacci numbers

I'm working on proving the problem that states $\text {The sequence}$ {$F_n$} $\text {is defined by the} \ F_1=F_2=1, F_{n+2}=F_{n+1}+F_n \ \text {for} \ n \ge 1.$ $\text {For any natural number m, ...
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2answers
24 views

Proving of Inequalities

How to prove: If $a>0$ and $b>0$, and $a^2>b^2$, then $a>b.$ I've tried different methods but I really can't prove this one. Thank you for your help!
10
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1answer
108 views

Prove that $a < b\sqrt{3}$ under conditions given

There are integers $a$ and $b$ such that: 1) $a > b > 1$ 2) $ab+1$ is divisible by $a+b$ and $ab-1$ is divisible by $a-b$. Prove that $a < b\sqrt{3}$. It's really hard, do you see a ...
2
votes
1answer
208 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
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2answers
54 views

Is this proof by induction for a sum of odd squares correct?

Statement: $1^2 + 3^2 + 5^2 + ... + (2n - 1)^2 = (n/3)*(2n-1)*(2n+1)$ Proof by induction -Base case: when $n = 1$ $1^2 = 1/3 * (2 * 1 -1) * (2 * 1 +1) = 1$ $1=1$ hence statement holds for $n = 1$ ...
1
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1answer
34 views

Proof of big-O notation

Prove the following: If f is a polynomial of degree $d$, then $f(n)=O(n^{d})$. For every $d \in N, n^{d} = O(e^{n})$ Intuitively, it makes sense to me that for the first one, growth order depends ...
0
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1answer
80 views

Convex n- sided polygon proof writing (homework question)

Would anyone be able to help me with the following problem or give me a push in the right direction? I am not entirely sure where to start and I have been looking at this problem for hours... Any help ...
0
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0answers
64 views

How do I prove that $f(n) + O(f(n)) = \Theta(f(n))$?

Here's what I have so far: $f(n) = \Theta(f(n))$ $C_1 f(n) < f(n) < C_2 f(n)$ $C_1f(n) + O(f(n)) < f(n) + O(f(n)) < C_2f(n) + O(f(n)) $ And then I run out of gas. The equal to sign ...
0
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1answer
37 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
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0answers
16 views

If $a,b,c$ are the vertices of a triangle in the complex plane, prove that the area of a triangle is $\frac{1}{2}|b-c|^2|Im\frac{c-a}{c-b}|$

I have trouble with this proof. I can get as far as the fact that we must position the vertex $c$ on the origin and then rotate by a factor of $|b-c|$. But then this gives: \begin{align*} ...
0
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2answers
25 views

Contrapositive proof question, is this a valid way

Definition: $a\in \Bbb Z$ is a perfect square if there is a $b\in\Bbb Z$ and $a = b^2$ To prove: if $m$ and $n$ are perfect squares, then $mn$ is a perfect square. I know that this can most easily ...
0
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1answer
37 views

Determining if two statements are equivalent, logical sense.

I am confused, I am working with proofs and I have the following statement to work with $\forall n\in\mathbb{N},P(n) \implies P(n+1)$ I have a second statement $\forall n\in\mathbb{N}, ...
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1answer
30 views

Suppose n is an integer. Use a proof by contrapositive to show if n^3 is even, then n is even

So, we assume that n is not even. Then, $n$ is odd, so $n= 2k+1$ for some integer $k$. Then, $(2k+1)^3 = 8x^3+12k^2+6k+1$. Would it be legal, then, for me to say $(8k^3+12k^2+6k)+1 = ...
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2answers
26 views

Suppose that x is an integer. Use a proof by contrapositive to prove that if 5x+7 is even, then x is odd.

I know that we assume x is even. So, as x is even, x = 2k for some integer k. Then, that would make for 5(2k)+7 = 10k + 7. And this is where I'm stuck. I know that it isn't complete at 10k+7 to ...
2
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2answers
47 views

Where should I place the notorious '+c'?

Consider the following proof - $$I=\int \sin (\ln x)dx\\\iff I=\sin(\ln x)x-\int\frac{ \cos (\ln x) }{x}\cdot {x} dx \\\iff I=x\sin (\ln x)-\int\cos(\ln x)dx\\\iff I=x\sin(\ln x )-[x\cos(\ln ...
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1answer
26 views

Is this correct structural induction (subsets)?

The last question (which is always the hardest) of my Induction Exercises goes like this: Let S be the subset of $\mathbb{Z}$ defined by: -12,20 $\in$ S if x,y $\in$ S, then x+y $\in$ S We use ...
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2answers
25 views

Understanding and writing limit proofs

I got this question : Consider $a_n,\:b_{n\:}$ sequences such that for every n , $0\le a_n\le \:b_{n\:}$. Let $\lim _{n\to \infty }\left(\frac{b_n}{a_n}\right)\:=\:1$, and $a_n$ is a ...
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0answers
10 views

Cartesian product proof with counterexample

I was asked to disprove the following statement by counterexample: Let A, B and C be sets. If A x C = B x C then A = B I was under the impression that: (x1, y1) = (x2, y2) if and only if x1 = x2 and ...
0
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1answer
14 views

Proving order of magnitude

Generally how much proof must be given to prove a statement of order-of-magnitude? for example: $n^2 + 2 log (n) = O(n^2)$ $2 log (n)$ has a lower order of magnitude than $n^2$ so it can be argued ...
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2answers
37 views

Proving by induction, if the base case fails to meet the main condition, what do we do?

I have to determine the number $x$ of subsets with odd cardinalities of a set $S$ and then prove that I'm correct. I determined the number $x$ is obtained using the formula $2^{n-1}$ where $|S| = n$. ...
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2answers
26 views

Finding counterexamples and proving a transformation is linear

Can someone please explain instances where $ f^{-1} (f (A)) \not = A $ and $f (f^{-1}(B)) \not = B$ if $ f:X \rightarrow Y $ and $ A$is a subset of X and B is a subset of Y? I can't think of when this ...
2
votes
1answer
88 views

Euler proof of the formula involving factorial?

Let me be formal and write the formula Euler's Formula: Let $a$ and $n$ be nonnegative integers with $a\geq n.$ Then $$n!=a^n-\binom{n}{1}(a-1)^n+\binom{n}{2}(a-2)^n-\binom{n}{3}(a-3)^n+ > ...
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6answers
67 views

Proof that every number has at least one prime factor

Prove that for $ n \geq 2$, n has at least one prime factor. I'm trying to use induction. For n = 2, 2 = 1 x 2. For n > 2, n = n x 1, where 1 is a prime factor. Is this sufficient to prove the ...
2
votes
1answer
22 views

Congruence modulo proof

Statement: a = 1 (mod 5) then a^2 = 1 (mod 5) Direct proof (we also have the option of proving by contraposition): let a = 5m + 1 then a^2 = (5m+1)^2 = 25m^2 + 10m + 1 = 5m(5m + 2) + 1 hence a^2 ...
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0answers
28 views

Proving that a value is a multiple of 11

I need to prove that 12**n - 1 is a multiple of 11 for every value of n (part of N). This is clearly a proof-by-induction problem. My base case is 0, where I assume n = 0 will give a resulting 0 ...
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2answers
53 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos ...
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0answers
29 views

How to write a formal proof of the statement: For all integers p, m, n, if p|m and p|n then p|(m+n)

Prove: For all integers $p$, $m$, $n$, if $p|m$ and $p|n$ then $p|(m+n)$ Proof: Let $p,m,n \in \mathbb{Z}$. Suppose $p|m$ and $p|n$. Then $\exists x,y\in \mathbb{Z}$ such that $m = px$ and $n = ...
0
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4answers
45 views

How to write a formal proof of the statement: For all integers n, if n is a multiple of 5 then 3n is a multiple of 5.

Prove: For all integers $n$, if $n$ is a multiple of $5$ then $3n$ is a multiple of $5$. Proof: Assume $n$ is a multiple of $5$. Then $n$ must have the form $5k$ where $k \in \mathbb{Z}$. We have ...
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3answers
42 views

Correctly negating “there exists a subset of $S$ that is a basis for $V$”

I would like to prove the following by contradiction: "Let $V$ be a vector space having dimension $n$, and let $S$ be a subset of $V$ that generates $V$. Prove that there is a subset of $S$ that is ...
0
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4answers
46 views

Prove that $2|(x^4-3) <=> 4|(x^2+3)$

Prove that $2|(x^4-3) <=> 4|(x^2+3)$ What i have right now is: Consider the case (=>): Since $x^4-3$ divides $2$ then, there must exist n belongs to integer, such that $n = \frac{x^4-3}{2}$ I ...
0
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2answers
24 views

Limit of |x-2| as x approaches -2

I believe that it equals -4. In the epsilon-delta definition, we can set delta equal epsilon and I become this satisfies the definition. The problem is I can't seem to prove based on this that 0 less ...