Let d be a metric on X. The distance between an element p E X and a nonempt; subset AC X is defined by d(p, A) = inf{d(p, a)la E A}. Similarly, the distance between two subsets A, BC X is defined by d(A, B) = inf{d(a, b)|a E A, b e B}. (a) Show that if An B+0, then d(A, B) = 0. (b) Show: Let B be a nonempty subset of X. If x, y E X, then |d(x, B) – d(y, b)| < d(x, y).

Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Let d be a metric on X. The distance between an element p E X and a nonempty
subset AC X is defined by
d(p, A) = inf{d(p, a)la E A}.
Similarly, the distance between two subsets A, BC X is defined by
d(A, B) = inf{d(a, b)|a E A, b e B}.
(a) Show that if An B#0, then d(A, B) = 0.
(b) Show: Let B be a nonempty subset of X. If x, y E X, then
|d(x, B) – d(y, b)| < d(x, y).
Transcribed Image Text:Let d be a metric on X. The distance between an element p E X and a nonempty subset AC X is defined by d(p, A) = inf{d(p, a)la E A}. Similarly, the distance between two subsets A, BC X is defined by d(A, B) = inf{d(a, b)|a E A, b e B}. (a) Show that if An B#0, then d(A, B) = 0. (b) Show: Let B be a nonempty subset of X. If x, y E X, then |d(x, B) – d(y, b)| < d(x, y).
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