# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $u/v \ge z/w$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, $\mid B \mid^{\mid A \mid}$ ...
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### 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 ...
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### 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 ...
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 ...