DWe1
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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 ...

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. ...

$100$ 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 ...

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. ...

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 ...

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 ...

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

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, $e^{|x|} \geq e^x = |e^x|$.

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 ...

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

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 ...

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 $n \in \... View answer 1 votes$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\}$... View answer 1 votes a matrix is invertible if and only if the determinant is nonzero. The determinant in this case is$6 - t^2$. View answer Accepted answer 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 ... View answer Accepted answer 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 ... View answer 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$...

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 $b_1 - ... View answer Accepted answer 1 votes We have the assumption that an upper bound u of a nonempty set$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$.... View answer 1 votes 2) A function is periodical if there exists a$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))^...

$X_n \times X_n$ 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 ...

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 ...

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, $7^{16} \equiv 1 \mod 24$, ...
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 $\tilde{f}(x) = f(x - \... View answer Accepted answer 0 votes Your question about (vi): You apply the specific operation that defines the group everywhere, also to inverses, the group only 'knows' that operation. In this case, you want to reach the unit, which ... View answer 0 votes We have$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$3$. View answer 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 ... View answer 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 $78 = 0.1x - 20$.