This exercise demonstrates that if we drop either of the defining conditions for planar spatial regions...

   

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This exercise demonstrates that if we drop either of the defining conditions for planar spatial regions A and B, then IA, B (1, 1, 0, 0) need not imply A = B. (a) Find an example of regularly closed sets A and B in the plane such that = IA, B = (1, 1, 0, 0) and A B. - (b) Find an example of closed sets A and B in the plane, each having an interior that is an open ball, such that IA, B (1, 1, 0, 0) and A ‡ B. we prove that an open ball in R2 is connected, meaning it cannot be expressed as the union of two disjoint nonempty open subsets. Therefore a set with an interior that is an open ball satisfies the second condition to be a planar spatial region.) This exercise demonstrates that if we drop either of the defining conditions for planar spatial regions A and B, then IA, B (1, 1, 0, 0) need not imply A = B. (a) Find an example of regularly closed sets A and B in the plane such that = IA, B = (1, 1, 0, 0) and A B. - (b) Find an example of closed sets A and B in the plane, each having an interior that is an open ball, such that IA, B (1, 1, 0, 0) and A ‡ B. we prove that an open ball in R2 is connected, meaning it cannot be expressed as the union of two disjoint nonempty open subsets. Therefore a set with an interior that is an open ball satisfies the second condition to be a planar spatial region.)