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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0
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3answers
163 views

Using rules of inference (Leibniz) to prove theorems.

Leibniz: If $A \equiv B$ is a theorem, then so is $C[p:= A] \equiv C[p:= B]$. Note: p is "fresh" means p doesn't occur in $A, B, C$. I am trying to understand how to use Leibniz rule of inference for ...
0
votes
1answer
48 views

Critical Points and Gradients/Derivatives

Plot the function $f(x)= 3+\cos(3x)-0.5\sin(5x)+0.2\cos\left(10x-\left(\frac{\pi}{4}\right)\right)$. Estimate how many critical points are on the interval $[0,2\pi]$. Consider $\mathbb{R}^{20} \to ...
1
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4answers
93 views

Prove or Disprove the existence of a basis

I'm asked to prove or disprove the existence of a basis $(p_0,p_1,p_2,p_3)$ of $F(t)(3)$ (Polynomials of degree at most 3) such that each of the polynomials $p_0,p_1,p_2,p_3$ satisfies the equation ...
4
votes
0answers
74 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
0
votes
1answer
48 views

Union and Intersection of families of initial segments

I'm trying to show that unions and intersections of families of initial segments are initial segments. An initial segment of a partially ordered set X is a subset of A such that, for every x$\in$X ...
1
vote
1answer
41 views

Negate Implication Written as a Sentence without “If …, Then …” [Chartrand P246]

P246 Theorem 10.4: Every infinite subset of a denumerable set is denumerable. P252 Theorem 10.10: Let $A \subseteq B$ be sets. If $A$ is uncountable, then $B$ is uncountable. I'm aware how ...
0
votes
2answers
46 views

$|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$

Let $a,b,c ∈ \Bbb R$ Show that $|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$ Not sure where to start
3
votes
1answer
97 views

Prove! that $a+(1/a) ≥ 2$ and $a+(1/a) ≤2$

Let $a \in R$ If $a>0$, then $a+\frac1a\geq2$ If $a<0$, then $a+\frac1a\leq2$ This is how someone explained the first one to me but still not really sure about it. Proof: ...
0
votes
0answers
52 views

Injection or Surjection Proof

Show that, given sets X and Y, there is an injection or a surjection X -> Y. I have a plan to prove this but struggle with the small details and writing the actual proof. 1) I will assume that ...
0
votes
1answer
300 views

Help to clarify proof of Euler's Theorem on homogenous equations

Why is the last step (setting $\lambda = 1$ allowed? I have trouble accepting this because if I set $\lambda =1 $ at the very start, then: $f(\lambda x , \lambda y)=\lambda^r f(x,y)$ becomes ...
2
votes
1answer
57 views

Prove that if SupS = infinity then for every N > 0 there exists an element s of S such that s > N

How would I prove this? Would I use upper bounds or lower bounds or would I do a proof by contradiction?
1
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2answers
55 views

Injective Equivalence

I'm trying to prove that these two statements are equivalent. I've already proven that $f$ injective implies that $$f^{-1} \left(f(B)\right) = B$$ but I need to show that $$f^{-1} \left(f(B)\right) = ...
0
votes
1answer
48 views

Surjection Equivalence [duplicate]

I'm trying to show that if a function $f:X\to Y$ is surjective it's equivalent to saying that $f \left(f^{-1}(B)\right) = B$ for each $B \subseteq Y$. The definition of surjective that I'm using is ...
-1
votes
2answers
108 views

Totally Ordered Set successor and predecessor unique

I'm trying to prove that, in a totally ordered set, an element can have at most one successor and at most one predecessor. I know that if x < y and there is no z $\in$ X with x < z < y then ...
1
vote
3answers
63 views

How can you prove this equality?

I am trying to figure out the following equality, but cannot seem to get anywhere. I tried integrating by parts, but that blew up when I set u = (log x)^n and tried to take log (0). I also tried ...
0
votes
3answers
43 views

Equivalence Relation Statements Proof

Let $\sim$ be an equivalence relation on a class $X$. The following are equivalent for $x,y \in X$. 1) $[x]=[y]$ 2) $x \sim y$ 3) $x \in [y]$ 4) $y \in [x]$ 5) $[x] \bigcap [y] \neq \emptyset$ ...
0
votes
1answer
55 views

Transitive Class

Prove that every nonempty transitive class has $\emptyset$ as a member. A class is transitive if each of its members is a subset of it. i.e. if t $\in$ T then t $\subseteq$ T. This is what I have ...
0
votes
2answers
36 views

How should the sequence or list $k+1, k+2, \ldots, m$ be interpreted in a proof when $k \ge m$?

How should the sequence or list $k+1, k+2, \ldots, m$ be interpreted in a proof when $k \ge m$ ? Context: Suppose the matrix $K$ ($m \times i$) has $k$ pivots and let $q$ be the first column of the ...
0
votes
1answer
119 views

Unions and intersections of indexed families of transitive sets are transitive

Let $\{ T_a \}_{a \in A}$ be a family of transitive sets. Prove that $\bigcup_{a \in A} T_a$ and $\bigcap_{a \in A} T_a$ are transitive. Assume $A \neq \emptyset$. I'm not sure how to apply the ...
0
votes
3answers
78 views

Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$

Problem Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$. Please help with the above. I have no idea what's going on. An explanation would be nice.
3
votes
1answer
87 views

I want to figure out how many Topologies are in the set X

I have the set $$X = \{1, 2, 3\}$$ and I want to figure out how many different topologies I can get from the set $X$ so what I have done is assumed that the empty set and the whole set are in $T$ ...
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votes
3answers
69 views

Class Transitivity Proof

Prove that a class $T$ is transitive iff $\bigcup_{t \in T},\, t \subseteq T$ iff $a \in t$ whenever $a \in b$ and $b \in T$. I know that I need to begin by proving the first statement implies ...
2
votes
0answers
84 views

Induction: Show: $\frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times … \times \frac{2n-1}{2n} \leq \frac{1}{\sqrt{3n+1}}$ [duplicate]

The question: Show by using induction that: $\frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times ... \times \frac{2n-1}{2n} \leq \frac{1}{\sqrt{3n+1}}$ for all $n$ $\in$ $Z_+$ My attempt at a ...
4
votes
3answers
116 views

Proving that Spec(α) and Spec(ß) partition positive integers iff α and ß are irrational and 1/α + 1/ß = 1

From Concrete Math, problem 3.13 asks: "Let α and ß be positive real numbers. Prove that Spec(α) and Spec(ß) partition positive integers if and only if α and ß are irrational and 1/α + 1/ß = 1" The ...
1
vote
2answers
81 views

Set Theory and Equality

Let $A$ and $X$ be sets. Show that $X\setminus(X\setminus A)\subseteq A$, and that equality holds if and only if $A\subseteq X$. I understand why this holds but am not sure how to 'show' this. Any ...
3
votes
2answers
214 views

Stuck with a tricky existence proof

Show that there exists a continuous function $f: [-1, 1] \rightarrow \mathbb{R}$ such $f(0) = 1$ and $f(x) = \frac{2-x^2}{2} \cdot f(\frac{x^2}{2-x^2})$ $\forall x \in [-1, 1]$ I tried putting ...
-1
votes
1answer
47 views

Proof that two sets have the same cardinality.

Let J be the set of all even finite subsets of a set M, and U the set of the odd. Show that J and U have the same cardinality. To tell the truth, I haven't gotten far. I would appreciate any help!
0
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1answer
29 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
1
vote
4answers
94 views

Calculating $\dfrac{1}{2^x}$ using $5^x$

If we look at the decimal equivilents of $2^{-n}$, we see they resemble $5^n$ with a decimal point in front of them: $\begin{align} 2^{-1} &= 0.5 \\ 2^{-2} &= 0.25 \\ 2^{-3} &= 0.125 \\ ...
0
votes
1answer
26 views

Fundamental Matrix with Sums

Let $$\Phi(t)=\begin{bmatrix} x_{11}(t) & x_{12}(t)\\ x_{21}(t) & x_{22}(t) \end{bmatrix} $$ be a fundamental matrix for $$x'=A(t)$$ where $$A=\begin{bmatrix} a_{11}(t) & a_{12}(t)\\ ...
0
votes
1answer
96 views

Stuck on writing a proof.

I am taking a discrete math class, and am still really new to writing proofs. I was wondering if anyone could help me with a problem. I am pretty confused on what it is even asking. Here is the ...
4
votes
2answers
365 views

Definition: Theorem, Lemma, Proposition, Conjecture and Principle etc.

Definition: Theorem, Lemma, Proposition, Corollary, Postulate, Statement, Fact, Observation, Expression, Fact, Property, Conjecture and Principle Most of the time a mathematical statement is ...
4
votes
3answers
106 views

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction: How to prove one of them?

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction How to prove one of them ? On Proofwiki there is an article proving the equivalence of the ...
1
vote
1answer
117 views

Complex numbers - exponential numbers - (double angles?)

I am half stumped on this rather confusing problem: Let x, y, and z be real numbers such that $\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0$. Prove that $\cos 2x + \cos 2y + \cos 2z = \sin ...
3
votes
4answers
164 views

Complex numbers - Exponential numbers - Proof

Let $z$ be a complex number, and let $n$ be a positive integer such that $z^n = (z + 1)^n = 1$. Prove that $n$ is divisible by 6. For this problem I am stumped...how should I begin? Also there's a ...
0
votes
1answer
77 views

Why isn't this a sufficient proof?

So basically, we have a question that asks us to prove that given a particular Deterministic finite automaton (DFA), there is a symbol for which we can get to a state $q$ from a state $p$ given a ...
0
votes
1answer
112 views

Does Graphical evicence count as / contribute to a Proof in Mathematics?

Several questions such as the following have an answer with pictures in it. How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$ How prove this inequality ...
1
vote
2answers
68 views

What is wrong with the proof given below?

This problem comes from Solow's book, 2nd edition. What is wrong with the proof given below? If $r$ is a real number with $|r| \leq 1$, then for all integers $n \geq 1, 1 + r + r^2 + \ldots + ...
0
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1answer
177 views

Proof that the set of all invertible $n \times n$ matrices of real numbers is a vector space.

I'm studying Algebra and I'm asked to prove or disprove the statement above. I found that is true, but I'm not sure how to prove it. My problem here is that this statement is too "broad", i.e. I ...
0
votes
1answer
60 views

Prove by Hilbert deduction: ⊢∃x(AvB)→(∃xAv∃xB); ⊢(∃xAv∃xB)→∃x(AvB)

I'd really like your help proving: 1)⊢∃x(AvB)→(∃xAv∃xB) 2)⊢(∃xAv∃xB)→∃x(AvB) Our proof system contains next Hilbert's axioms: 1.A→(B→A) 2.(A→B)→((A→(B→X))→(A→X)) 3.(A&B)→A 4.(A&B)→B ...
2
votes
2answers
76 views

Upper bound of a set (equivalence problem)

I took a real analysis course three years ago and I unfortunately didn't get all of it, starting with basics. Question: Let $E$ be a set of real numbers. Show that $x$ is not an upper bound of $E$ ...
1
vote
2answers
209 views

Basic proof problem from “How to Prove it A Structured Approach”

I got the book How to Prove it A Structured Approach and I'm ashamed to admit I failed to even do the first problem in the introduction chapter: a) Factor $2^{15} - 1 = 32767$ into a product of two ...
1
vote
6answers
578 views

If $n$ is a positive integer and is not a perfect square

If $n$ is a positive integer and is not a perfect square, how do you prove that $n^{1/2}$ is irrational?
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2answers
82 views

Prove that set contains least element.

Let $A\not=\emptyset ,A\subset \mathbb{Z}$ and if $(\exists d\in \mathbb{Z})(\forall a\in A):d\le a$ then set A contains least element. How do I prove this? I understand I can use WOP principle. What ...
0
votes
3answers
108 views

Integrating $\int\sin^{-2}xdx$ [duplicate]

I am trying to prove that $$ \int\frac{1}{\sin^2(x)}dx = -\cot(x) + C $$ but I have difficulties, I don't know where to start, I can't substitute anything with $sin(x)$ because I don't have a $cos(x)$ ...
2
votes
0answers
71 views

Set of limit points of S is closed in a metric space X

A point $x \in X$ is a limit point of a subset S of X, if every ball $B(x;\varepsilon)$ contains infinitely many points of S. Show that x is a limit point of S iff there is a sequence {$x_{j}$} ...
0
votes
2answers
36 views

How to prove statement with two variable by induction

I am trying to prove following statement: $[m,n]$ is a set of functions defined as $f \in [m,n] \leftrightarrow f: \{1,...,m\} \rightarrow \{1,...,n\}$. The size of $[m,n]$ is $n^m$ for $m,n \in ...
6
votes
1answer
205 views

Intuition behind proof and verification partially ordered sets

Hi everyone in the book that I read I have trouble to understand the argument of the proof at the below proposition. There is a lot of point which are left as exercises, which is great. One of these ...
1
vote
1answer
62 views

How do i do this process *precisely*?

Let $[a,b)\times [c,d)$ be a rectangle $R$ in $\mathbb{R}^2$. Let $\{[u_k,v_k)\times [p_k,q_k)\}_{1≦k≦n}$ be a mutually disjoint finite sequence whose union is $R$. Then we can decompose this into ...
1
vote
2answers
64 views

complex numbers - Proving

Prove algebraically that $|w+z|\le|w|+|z|$ for any complex numbers w and z. This is what I got so far: Since $|z|^2 = z \overline{z}$, we square both sides: $ |w+z|^2 = (w+z) \overline{(w+z)} \le ...