First slide

Operations on Sets

Question

Let $A = \{(x, y) : x > 0, y > 0, x^{2} + y^{2} = 1\}$ and let $B = \{(x, y) : x > 0, y > 0, x^{6} + y^{6} = 1\}$ Then $A \cap B$

Moderate

A

$A$
B

$B$
C

$\emptyset$
D

${(0, 1), (1, 0)}$

Solution

$\begin{array}{l} \begin{array}{l} x^{6} + y^{6} = (x^{2} + y^{2}) (x^{4} - x^{2} y^{2} + y^{4}), so if (x, y) \\ A \cap B then 1 = x^{4} - x^{2} y^{2} + y^{4} = {(x^{2} + y^{2})}^{2} - 3 x^{2} y^{2} \\ = 1 - 3 x^{2} y^{2} \end{array} \\ \Rightarrow x^{2} y^{2} = 0 x = 0 or y = 0 but x > 0, y > 0 \\ so A \cap B = ϕ \end{array}$

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