0
votes
2answers
36 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
0
votes
5answers
49 views

Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

What I've got so far is this: Base case: n = 3 then $3 *2 * 1 = 6$ and $3^3 = 27$ $\therefore 6 < 27, 3! < 3^3$ So the base case is true. So if we assume $n! < n^n$ (n > 2) $(n + 1)! = ...
0
votes
0answers
19 views

Orthonormality and fourier transform

If $g\in\mathcal{L}^2(\mathbb{R})$ then $\sum_{k\in\mathbb{Z}} |\hat{g}(\zeta+2k\pi)|^2=1$ for a.e $\zeta\in \mathbb{R} \Rightarrow \{g(.-k): k\in \mathbb{Z}\}$ is an orthonormal system. Please ...
1
vote
2answers
87 views

Prove: If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

Can someone tell me if this proof is acceptable? Assume $A \not\subseteq C$. This means that there is an $x \in A$ such that $x \not\in C$. But since $\forall x \in A: x \in B$ and $\forall x \in B: ...
0
votes
2answers
23 views

Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...
2
votes
4answers
87 views

Prove $A \subset \emptyset \iff A = \emptyset$

How does one prove this? Can one prove by contradiction by saying: Let $A$ be any set such that $A$ contains at least one element. Now assume $A \subset \emptyset$. This is clearly absurd by the ...
0
votes
2answers
52 views

If a converse of an implication is false, does this mean that the proof of that implication will always have an implication that is not reversible?

Let $f:X \rightarrow Y$ be a function and $B_1, B_2 \in \mathcal{P}(Y)$. Prove that $B_1 \subseteq B_2 \Rightarrow \overleftarrow{f}(B_1) \subseteq\overleftarrow{f}(B_2)$. My attempt: $\begin{align} ...
2
votes
2answers
30 views

Help to prove $f$ is surjective $\Leftrightarrow \forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $

Let $f:X \rightarrow Y$ be a function with graph $G_f \subseteq X \times Y$. Prove that $f$ is surjective if and only if $\forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $ I have some ...
1
vote
0answers
57 views

Prove: $a^m\cdot a^n \cdot a^p=a^{m+n+p}$

How can I prove the following: Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove: 1) $a^0\cdot a^0 \cdot ...
0
votes
0answers
31 views

Trying to prove that two angles are congruent in a isosceles trapezoid

I was given this assignment to do the following. Write a paragraph proof for the following scenario. Given: KLMN is an isosceles trapezoid. Prove: ∠LKM is congruent to ∠MNL The thing is that I ...
2
votes
2answers
73 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
1
vote
2answers
52 views

Wondering if proof is proper

so I have been working on learning some new math in order to prepare for next year. I have been trying to learn proofs, and doing practice questions however the only problem is there are not answers. ...
4
votes
1answer
39 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...
2
votes
1answer
28 views

Question on Induction (Very Simple)

I've just started a course in mathematics at university, and our current topic is mathematical induction. I've been given the following question: $$1+4+4^2+....+4^{n-1}=\frac{4^{n}-1}{3}.$$ I get ...
2
votes
1answer
42 views

Proof Verification for Discrete Math Class

Prove that $n^2$ is even iff $n$ is even. I proved it like this: Case I: $n$ is even 1) $n = 2a$ $(a\in Z)$ 2) $n^2 = 4a^2 = 2(2a^2)$ 3) $2a^2 = K$ $(K \in Z)$ 4) $n^2 = 2K$ Case II: $n$ is ...
0
votes
1answer
25 views

Proof Writing Help: $P_UT=TP_U \Leftrightarrow U$ and $U^{\perp}$ are $T$-Invariant

I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my ...
2
votes
2answers
60 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
1
vote
2answers
43 views

Is there a direct proof of the following?

I have been warned by my Lecture as well as several other sources that while proof by contradiction is useful and is certainly needed in some cases, it is often overused. In a effort to learn, I ...
1
vote
3answers
148 views

Proposed proof of set theoretic result

I am tasked with proving the following: $$ (A - B)\cap (B-A) = \varnothing $$ My Attempt: Suppose there exist a $x \in (A - B)\cap (B-A) $ then: \begin{align*} x \in (A - B)\cap (B-A) &\iff ...
1
vote
1answer
50 views

How to prove a very basic algorithm by induction

I just studied proofs by induction on a math book here and everything is neat and funny: the general strategy is to assume the LHS to be true, and use it to prove the RHS (for the inductive step). Now ...
1
vote
1answer
23 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
1
vote
0answers
32 views

Proof Validation Function From Integers to Rationals is Continuous

I am teaching myself real analysis, so any help is greatly appreciated. Let the function be defined as $F : Z \rightarrow Q$ where $Z$ is the set of integers and $Q$ is the set of rational numbers, ...
2
votes
0answers
35 views

Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 ⊆ A$, $B_2 ⊆ A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 ⊆ B_2$ then ...
3
votes
1answer
50 views

$f'$ strictly increases and $f'(c)=0$. There exist $x_1 < c < x_2$ such that $f'(c)=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$

Question: Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Assume that $f'$ is strictly increasing. Show that for any $c\in(a,b)$ such that $f'(c)=0$, there exist $x_1, x_2 \in [a,b], ...
2
votes
0answers
61 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
1
vote
1answer
24 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
6
votes
1answer
98 views

Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...
0
votes
1answer
32 views

double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
2
votes
2answers
21 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
0
votes
0answers
40 views

Mathematical logic and proofs involving absolute values

Is the following proof correct? Let ($\forall$ x, y $\in $$\Bbb R)$ $|x-y| \le |x| +|y|$ case#1: Suppose x and y $\ge0$. We want to show that $|x-y| \le |x| + |y|$. Since $|x|\ge x$ and $|y|\ge y$, ...
0
votes
1answer
33 views

Are these proofs correct?

I haven't formally learn how to do proofs, but I attempted some of these. It'd be great if you guys can check them and give me some pointers. Thanks!
2
votes
1answer
23 views

Proof-Writing $\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$

At the end of this message there are two steps that I do not understand. The proof wants to show in the end that : *$\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$ by definition we have ...
2
votes
0answers
74 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
2
votes
2answers
53 views

Logic Proof with Natural Deduction: if I assume the antecedent, do I still have to prove the consequent?

I have the unpleasent feeling that my "proof" is dead wrong. The core of my concerns is: when I have something like A -> (B -> C) and I assumed ...
1
vote
1answer
27 views

Let $A =[a,b,c,f,g,i], B=[b,f,h]$ and $ C = [a,k,l,m]$ Show that $\backslash$ is not associative

Question : Let $A =[a,b,c,f,g,i], B=[b,f,h]$ and $ C = [a,k,l,m]$ Show that $\backslash$ is not associative by comparing $(A \backslash B) \backslash C$ with the set $A \backslash(B \backslash C)$. ...
1
vote
1answer
45 views

Question about Logic Proof

Assuming $P$ is a unary predicate and $Q$ is a propositional variable, I'm trying to prove the following implication: $$ (\forall x (P(x)\rightarrow Q)\rightarrow ((\forall x P(x) )\rightarrow Q) $$ ...
2
votes
6answers
76 views

Prove if $ab>0$ and $bc<0$ then $ax^2 + bx +c = 0$ has two real solutions

For there to be two real solutions in a quadratic equation the discriminant, $b^2-4ac$, has to be positive, so $b^2-4ac > 0$. Rearranging the equation, I get $b^2 > 4ac$. Then $b > ...
0
votes
3answers
41 views

Let $ n,m \in Z$. Prove the following: If $m$ and $n$ are even, then so is $mn.$

Question: Let $ n,m \in Z$. Prove the following: If both $m$ and $n$ are even, then so is $mn.$ Attempt: By Definition 2.3.1, we let $ n \in Z$. Then, $n$ is even whenever there exists some $ k \in ...
1
vote
0answers
27 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
0
votes
1answer
18 views

Struggling with proof, by contrapositive?

I am having trouble solving this proof. I tried to do a proof by contrapositive. Q = $(u+z)/(v+w) < z/w$ P = $(u/v < x/y \land x/y < z/w)$ Assuming $\lnot Q$ got me: $u/v \ge z/w$ ...
1
vote
1answer
43 views

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W This is something from a practice sheet I got. I'm studying for a linear algebra final. I am unsure if we have ...
2
votes
1answer
34 views

Doubt on proof of Implicit function theorem

On The second part of the proof, where it's stated that V is open as it is the inverse image of the open set $V_0$ under the continuous mapping $y \rightarrow (0, y)$. Let $\pi$ be this continuous ...
2
votes
2answers
68 views

Proving corollary to Euler's formula by induction

I'm currently looking at two proofs to the following corollary to Euler's formula and I'm not quite seeing how the authors can make a specific assumption in their proof. One proof comes from my ...
1
vote
1answer
77 views

Prove that if graph $G$ is a 3-connected planar graph then its dual must be simple.

I'm trying to study for a quiz. I think I'm on the right track with this problem, however, I'm having a difficult time formalizing it. Prove that if graph $G$ is a 3-connected planar graph then its ...
0
votes
2answers
46 views

Show that $A^{(x,y)}$ is countable.

Question: Let $A$ be a countable set $A^{(x,y)}$ the set of all functions from $(x,y)$ to $A$. Show that $A^{(x,y)}$ is countable. My attempt: By proposition 7.1.2iii, $\mid B \mid^{\mid A \mid}$ ...
0
votes
1answer
28 views

The Open Set $X-\lbrace x \rbrace$

I am task with proving the following: if $x \in X$ then $X- \lbrace x \rbrace $ is an open set I kind of have an idea but I am unsure about it and how to express it. I was thinking about using the ...
6
votes
0answers
67 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
2
votes
1answer
156 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 ...
1
vote
1answer
57 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...