Proving Subset Inclusion for Intersection of Function Images

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Homework Statement



Suppose f: A → B and that E,F are subsets of A.
Prove the following:
a) f(E \cup F) \equiv f(E) \cup f(F)
b) f(E \cap F) \subset f(E)\cap f(F)

Homework Equations



The Attempt at a Solution


So far I have solved the first one, but I am having trouble with the second. I have no idea where to begin.

BiP
 
Last edited:
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I don't believe b) is true..
 
Woops! Sorry I wrote it wrong. I'll change that.

BiP
 
Ok you that should just be a straight element proof then, just follow your nose, if b is in f(E∩F) then there exists an a in E∩F such f(a)=b, if a is in E∩F then a is in E and F.. and so on and so forth.
 
What does "and so on and so forth" supposed to mean? I don't understand your proof sorry. It's incomplete.

BiP
 
Bipolarity said:
It's incomplete.

Yes, because you are supposed to finish it.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

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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