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Question Number 155729 by henderson last updated on 03/Oct/21

A={(a,b)∈IR^2  / a^2 +b^2 ≤1}  prove that A can′t be written as the cartesian  product of two parts of IR.

$$\mathrm{A}=\left\{\left({a},{b}\right)\in\mathrm{IR}^{\mathrm{2}} \:/\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} \leqslant\mathrm{1}\right\} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{A}\:\mathrm{can}'\mathrm{t}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{the}\:\mathrm{cartesian} \\ $$$$\mathrm{product}\:\mathrm{of}\:\mathrm{two}\:\mathrm{parts}\:\mathrm{of}\:\mathrm{IR}. \\ $$

Answered by Kamel last updated on 04/Oct/21

A={(a,b)∈R^2 / −1≤a≤1 , −(√(1−a^2 ))≤b≤(√(1−a^2 ))}

$${A}=\left\{\left({a},{b}\right)\in\mathbb{R}^{\mathrm{2}} /\:−\mathrm{1}\leqslant{a}\leqslant\mathrm{1}\:,\:−\sqrt{\mathrm{1}−{a}^{\mathrm{2}} }\leqslant{b}\leqslant\sqrt{\mathrm{1}−{a}^{\mathrm{2}} }\right\} \\ $$

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