Proving Set Theory Union in Cartesian Products

rallycar18
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Homework Statement



Suppose A,B,C are sets. Prove that

A× (B U C)= (AxB) U (C x A)
 
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Have you tried the usual inclusion both ways method?
 
VeeEight said:
Have you tried the usual inclusion both ways method?

I'm not familiar..
 
Well if A is contained in B and B is contained in A, then A=B.
 
assume x \in A \times (B \cup C). Think about the definitions of Cartesian product and union: what can you conclude about the element x; can you use this information to show x \in (A \times B) \cup (A \times C)?
 

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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