Help with a simple group theory question please

Ineedhelpimbadatphys
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Homework Statement
the question is about topology, but i just want to know.

isn't {∅,R}∪{]a,∞[:a∈R} equal to {∅,R}
since every member of {]a,∞[:a∈R} is a real number?

or am i just completely misunderstanding unions and intersections?
Relevant Equations
above
above
 
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Ineedhelpimbadatphys said:
Homework Statement: the question is about topology, but i just want to know.

isn't {∅,R}∪{]a,∞[:a∈R} equal to {∅,R}
since every member of {]a,∞[:a∈R} is a real number?

or am i just completely misunderstanding unions and intersections?
Relevant Equations: above

above
Or is true. The elements of your sets are sets again. ##\emptyset\, , \,\mathbb{R}\, , \,\{r\,|\,r>a\}## are three sets, but here we consider them as the elements of ##\{\emptyset\, , \,\mathbb{R}\}## and ##\{(a,\infty )\}##. This makes the union a set with three elements, ##\emptyset\, , \,\mathbb{R}\, , \,\{r\,|\,r>a\}##.
 
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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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