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### The proof of Lemma 10.1 in Nik’s book about forcing

I’m an undergraduate student trying to teach myself set-theory. And I have some trouble understanding the density of a constructed set. In Lemma 10.1 of Nik’s book, it states: Let $G$ be a generic ...
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### Inequality involving sums with binomial coefficient

I am trying to show upper- and lower-bounds on $$\frac{1}{2^n}\sum_{i=0}^n\binom{n}{i}\min(i, n-i)$$ (where $n\geq 1$) in order to show that it basically grows as $O(n)$. The upper-bound is easy to ...
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### Explaining Conditional Probability and the Urn problem with white and black balls

I am trying to interpret the meaning behind the conditional probabilities when setting up the problem. Can someone confirm with me if I am understanding the problem correctly? Problem Statement: There ...
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### Finding a spanning set of a null space

$$A= \begin{bmatrix} -3& 6 &-1& 1 &-7\\ 1 &-2& 2& 3&-1\\ 2&-4& 5& 8& -4 \end{bmatrix}$$ Please I have a problem finding the spanning set of a null ...
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### Can imaginary i and -i be the elements of the group Z_2?

We know that the elements of the group Z_2 are -1 and 1. Can imaginary i and -i be considered the elements of the group $Z_2$?
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### The integral of $\frac{\sin(x^2)}{x}\, \text{d}x$ [duplicate]

I need to calculate the following integral; $$\int\limits_0^{∞} \frac{\sin(x^2)}{x}\,\text{d}x$$ But, I don't really have any ideas. Maybe using a new variable or integrating by parts would work. Can ...
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I am trying to do the following question: Determine whether each of the following line integrals is independent of path. If it is, find a function $h$ such that $d h=P d x+Q d y$. If it is not, find a ...
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### Complex number Method verification for finding the sum of $\arctan 1+\arctan2 + \arctan3$

In the complex number method posted by a fellow MSE member here : https://math.stackexchange.com/a/272244/ , I think its not true always as what he/she did , assuming $\operatorname{Arg}$ represents ...
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Suppose $Z_i$ is independent standard normal distributions, i.e. $Z_i\sim N(0,1)$, $i=1,2,\cdots, d$. What is the distribution of $$\sum_{i=1}^d (a_iZ_i+b_iZ_i^2).$$ I know when $a_i=0$, it is the ...