- #1
RJLiberator
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Homework Statement
Find the complement C' of the set C with respect to the space U if:
1. U = {(x,y,z) : x^2+y^2+z^2 ≤1}, C = {(x,y,z) : x^2+y^2+z^2 = 1}
2. U = {(x,y) : |x| + |y| ≤ 2}, C = {(x,y) : x^2 + y^2 < 2}
Homework Equations
Definition of complement: The elements of the space that do not exist inside the set.
The Attempt at a Solution
I wanted to reassure myself that I am doing things right.
For 1: My answer is C' = {(x,y,z): x^2+y^2+z^2 < 1}
However, why is this right? Couldn't I have (sqrt(1/3), sqrt(1/3), sqrt(1/3)) as (x,y,z) for the set C = 1, and then have C' = (sqrt(1/3), sqrt(1/3), 0) and then we have x and y equal in both C and C' ? Is my answer wrong or what am I missing in the understanding here?
For 2: x^2+y^2 < 2 is a circle with radius sqrt(2) where as |x| + |y| ≤ 2 is a square with corners on each x = - 2, x = 2, y= - 2, y = 2. So the area i am looking for is |x| + |y| - x^2 + y^2 ≤ 2.
But I am not sure how to put this into a final answer.
I can visualize it via the area, but it's hard to put this into an answer.