Abstract math, sets and logic proof

beatka6
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Homework Statement


If A is a set that contains a finite number of elements, we say A is a finite set. If
A is a finite set, we write |A| to denote the number of elements in the set A. We
also write |B| < ∞ to indicate that B is a finite set. Denote the sets X and Y by
X = {T : T is a proper subset of P(Z) or |T| < ∞}; Y = {T element of X : T≠ ∅}
Prove or disprove the following:
(there exist X element of R)(∅ element of R and ( for all S element of Y)(|R|≤ |S|}


Homework Equations




The Attempt at a Solution


I think that statement is true because of or in the statement, but I have no idea how to prove it
 
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I can't understand what it is that you are trying to show. Can you write it out in words?
 
It s number 5 from the attachment.
 

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Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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