Answer to Question #215074 in Calculus for Sarita bartwal

Question #215074
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
1
Expert's answer
2021-07-08T13:57:36-0400

Let us show that SC.S\subset C. If P(x,y,z)S,P(x,y,z)\in S, then x2+y2+z2<1.x^2+y^2+z^2<1. Since x20x^2\ge 0 and y2+z20,y^2+z^2\ge 0, the inequality x2+y2+z2<1x^2+y^2+z^2<1 implies x2<1.x^2<1. It follows that 1<x<1.-1<x<1. By analogy, y20y^2\ge 0, x2+z20x^2+z^2\ge 0 and x2+y2+z2<1x^2+y^2+z^2<1 imply y2<1,y^2<1, and hence 1<y<1.-1<y<1. In the same way, 1<z<1.-1<z<1. We conclude that P(x,y,z)C,P(x,y,z)\in C, and hence SC.S\subset C.



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