DWe1
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Prove or Disprove that there exists an integer $n$ such that $4n^2 -12n +8$ is prime
5 votes

Since $4$ divides $4n^2 - 12n + 8$, its prime factorization already has the factors $2^2$. There exists no number that is prime and has more than one prime factor. Hence, there cannot be a prime ...

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onto homomorphic mapping from $Z_4 $ to $Z_2 \times Z_2$
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4 votes

Both groups are finite and of the same size, hence surjectivity implies injectivity using the pigeonhole principle. Hence, requiring a homomorphism that is surjective actually requires an isomorphism. ...

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Factors and primes
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3 votes

$100$ is the LCM of $20$ and $n$. Hence, $100$ must be a multiple of $n$, so we only need to look at divisors of $100$ as possible values of $n$. Furthermore, divisors of $20$ will lead to $20$ as ...

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Do quotas lead to lower overall value assuming input types are equally distributed
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2 votes

This is not a mathematical question. The assumptions that you need to say such a thing are 'hidden'. If you look for a mathematical "proof" of a sociological problem, you're going to have a bad time. ...

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If a sequence converges, and a subsequence converges to $x$, does the sequence converge to $x$?
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2 votes

Yes, this is true. The sequence converges, so suppose it does not converge to $x$, then it converges to a $y \neq x$. Now, apply the theorem you mention to conclude that the subsequence should ...

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Why does small change in inputs for continous function cause only a small change in the outputs?
2 votes

Continuous functions are defined that way. You cannot prove it, itś simply what we define a continuous function as. We don't define a continuous function like you say you "know" it, "without any jumps ...

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Logic of the implication in $ε$-$δ$ proofs
2 votes

Let's just run through the proof. We want to prove that, for any $\epsilon > 0$, there exists $\delta > 0$ such that for all $x$, $0 < |x-2| < \delta \implies |(3x-1)-5|<\epsilon$. ...

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What properties of the exponential function causes $\mid e^{f(x)}| \leq e^{\mid f(x)\mid}$ to be true?
2 votes

For any $x \in \mathbb{R}$, we have $e^x \geq 0$, hence $|e^x| = e^x$. Also, $e^x$ is always increasing, and $|x| \geq x$. Therefore, $e^{|x|} \geq e^x = |e^x|$.

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Bounded functions
2 votes

The function is unbounded. An intuitive way of seeing this is to consider $t$ to be just below $-1$. Then, $t+1$ will be just below $0$, but we can get arbitrarily close. To formalize this idea, we ...

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An element of a subgroup's coset with one of its elements is itself ?
1 votes

$\implies$: Multiply $x$ with the unit element of $G$. $\impliedby$: Use that $H$ is closed under the operation.

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Quotient group concerning $\mathbb Z / 24 \mathbb Z$
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1 votes

If we view $G = \mathbb{Z}/24\mathbb{Z}$, we talk about the quotient group of the additive group $\mathbb{Z}$ and a subgroup $24\mathbb{Z}$. Hence, we "mod out" $24$ and set it equivalent to $0$. If ...

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Prove Fibonacci by induction using matrices
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1 votes

Take $n = 0$ and see that $$\begin{bmatrix} f(n) \\ f(n+1) \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ is clearly true: We just see the first two terms. Suppose for a particular $n \in \...

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Difference between subset and belong
1 votes

$A$ is a subset of $B$ means that every element of $A$ is also an element in $B$. $x$ belongs to $A$ if $x$ is an element of $A$ itself. Example: $A = \{1, 2, 3\}$ is a subset of $B = \{1, 2, 3, 4\}$...

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Invertible Matrices with t
1 votes

a matrix is invertible if and only if the determinant is nonzero. The determinant in this case is $6 - t^2$.

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About the proof of square root of 2 is irrational
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1 votes

A rational number could be written as a fraction of two integers that are not relatively prime. However, we can write a rational number as a fraction of two relatively prime integers. Hence, if we ...

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Applications on derivative
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1 votes

the derivative of the curve, $\frac{dy}{dx} = 6x^2$. For $x=2$, the slope of the curve is equal to $24$. For $x=-2$ the slope of the curve is also equal to $24$. Hence, the tangents on the curve will ...

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Proving (A∖B)∩C ⊂ A∖(B∩C)
1 votes

... hence $x \in A$, $x \not\in B$, $x \in C$. Since $x \not\in B$, $x$ is not in any subset of $B$, hence $x \not\in B \cap C \subset B$. We conclude that $x \in A$ and $x \not\in B \cap C$, hence $...

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cauchy sequence and least upper bound property
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1 votes

The simplest answer is using the monotone convergence theorem: If a sequence is bounded and monotone (only increases or decreases), it will be convergent. The sequence $b_n - a_n$ is bounded by $b_1 - ...

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Question on supremum proof
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1 votes

We have the assumption that an upper bound u of a nonempty set $S$ in $\mathbb{R}$ is the supremum of $S$ iff for every $ε>0$ there exists an $s_\epsilon \in S$ such that $u−\epsilon<s_\epsilon$....

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Periodicity of combination of trigonometric functions and more
1 votes

2) A function is periodical if there exists a $P$ such that for any real number $x$, $f(x) = f(x+P)$ Note that if $f$ is periodical, $f(x) = f(x+P)$ for any real number $x$, so it follows that $(f(x))^...

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Comparison for two finite sets.
1 votes

$X_n \times X_n$ is a collection of pairs of elements of $X_n$. $U$ is the subset of $X_n \times X_n$ where $i < j$. This means that the first element of the pair must be strictly smaller than the ...

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Is it true to say that any matrix in row echelon form, with columns as vectors, has linearly independent vectors?
Accepted answer
1 votes

If you want to know whether all column vectors are linearly independent, then no, $$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$ is an example of a ...

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How to use the totient function in the context of clocks?
0 votes

Hint: Euler's Theorem states that if $a$ and $n$ are coprime, then $a^{\varphi(n)} \equiv 1 \mod n$. Since $7$ and $24$ are coprime, we have $7^8 \equiv 1 \mod 24$. Hence, $7^{16} \equiv 1 \mod 24$, ...

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Translate on the horizontal axis the graph of a second degree polynomial
0 votes

If you want to translate $f$ with $\alpha$ along the horizontal axis, you want to find what the value $f(x)$ at the point $x + \alpha$, so your translation $\tilde{f}$ must be $\tilde{f}(x) = f(x - \...

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A few questions about determining if these sets are groups or not
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0 votes

Your question about (vi): You apply the specific operation that defines the group everywhere, also to inverses, the group only 'knows' that operation. In this case, you want to reach the unit, which ...

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Arithmetic Progression. Find 1st term and common difference
0 votes

We have $a_7 = 15$ and $\frac{7(a_1 + a_7)}{2} = 42$. By plugging in $a_7$ into the second equation, we get $a_1 = -3$. It follows clearly that the difference between two terms will be $3$.

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How $f(x)\le 0 \wedge f''(x)\ge 0 \implies f'(x)=0$?
0 votes

Sketch of proof: Suppose, for a certain $q \in \mathbb{R}$, $f'(q) = a > 0$. Then, since $f''(q) \geq 0$, $f'(x) \geq a > 0$ for all $x \geq q$. (If it would go to $0$ again, $f'(x)$ had to ...

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how much did the trip cost the school
0 votes

The 10% discount is given after a $\$20$ processing fee. So $78 = (x-20)*0.1 = 0.1x - 2$. What you said is equivalent to $78 = 0.1x - 20$.

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