Show that the set difference metric given by d(A, B) = size(A ? B) + size(B ? A) satisfies the metric axioms given on page 70. A and B are sets and A ? B is the set difference.
1(a). Because the size of a set is greater than or equal to 0, d(x, y) ? 0.
1(b). if A = B, then A ? B = B ? A = empty set and thus d(x, y)=0
2. d(A, B) = size(A?B)+size(B?A) = size(B?A)+size(A?B) = d(B,A)
3. First, note that d(A, B) = size(A) + size(B) ? 2size(A ? B).
? d(A, B)+d(B,C) = size(A)+size(C)+2size(B)?2size(A?B)?2size(B?
C)
Since size(A ? B) ? size(B) and size(B ? C) ? size(B),
d(A, B) + d(B,C) ? size(A) + size(C)+2size(B) ? 2size(B) = size(A) +
size(C) ? size(A) + size(C) ? 2size(A ? C) = d(A, C)
? d(A, C) ? d(A, B) + d(B,C)
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