Answer to Question #215074 in Calculus for Sarita bartwal

Question

Suppose S and C are subset of R^3. S is the unit open sphere with centre at the origin and C is the open cube= { P(x,y,z)| -1<x<1, -1<y<1, -1<z<1. Which of the following is true. Justify your answer
1. S⊂C 2. C⊂S

Answer 1

Let us show that $S\subset C.$ If $P(x,y,z)\in S,$ then $x^2+y^2+z^2<1.$ Since $x^2\ge 0$ and $y^2+z^2\ge 0,$ the inequality $x^2+y^2+z^2<1$ implies $x^2<1.$ It follows that $-1<x<1.$ By analogy, $y^2\ge 0$, $x^2+z^2\ge 0$ and $x^2+y^2+z^2<1$ imply $y^2<1,$ and hence $-1<y<1.$ In the same way, $-1<z<1.$ We conclude that $P(x,y,z)\in C,$ and hence $S\subset C.$