# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### Truth table of proof by contradiction

The following is the truth table for an implication: $(T\Rightarrow T) = T$ $(T\Rightarrow F) = F$ $(F\Rightarrow T) = T$ $(F\Rightarrow F) = T$ Now, in an implication involved in a proof by ...
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### Show that the following are equivalent:

If $f$ is a continuous function on a bounded set $S$, show that the following are equivalent: (a) the function $f$ is uniformly continuous on $S$. (b) it is possible to extend $f$ to a ...
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### Let $(s_n)$ be a sequence that converges…

Exercise 8.9 from Elementary Analysis: The Theory of Calculus by Kenneth A. Ross: Let $(s_n)$ be a sequence that converges. (a) Show that if $s_n \geq a$ for all but finitely many $n$, then ...
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### Changes in singular values of matrix when rows are added

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is: Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...
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### Prove or disprove $n \geq 2 ~\rightarrow~ \prod \limits_{i=1}^{n} \left ( 1 - \frac{1}{i^2} \right ) ~=~ \frac{n+1}{2n}$

I am working on one of my HW assignments $$\forall n \in \mathbb{Z}, ~ n \geq 2 ~\rightarrow~ \prod \limits_{i=1}^{n} \left ( 1 - \frac{1}{i^2} \right ) ~=~ \frac{n+1}{2n}$$ And i am not ...
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### Homomorphism Transitivity

Show that $G\times H\cong H \times G$ and that if $A\cong G$ and $B\cong H$, then $A\times B\cong G\times H$. So you make a isomorphism $\phi$ from $G \times H \mapsto H \times G$ and because this is ...
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### Prove: There exists an even integer $n$ that can be written in two different ways as a sum of two distinct primes.

I am working on this problem, Prove: There exists an even integer $n$ that can be written in two different ways as a sum of two distinct primes. I know: $3+13=11+5=16$ ...
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### Prove if A and B are n x n upper triangular matrices, so is AB

I'm trying to practice proofs for my linear algebra final and I've been stuck on this one for some time. I have $AB = [A\mathbf{b_1} \ A\mathbf{b_2} \ \dots \ A\mathbf{b_n}]$. I can show that ...
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### Critique my elementary proof for a set bounded above

Let $A$ and $B$ be two non-empty subsets of $\mathbb{R}$ that are both bounded above. $(i)$Prove that $A ∪ B$ is bounded above and prove $(ii)$ that $\sup(A ∪ B) = \max(\sup(A),\sup(B))$. for ...