problem : Let F(S) be the set of all finite subsets of a set S. For all A,B ∈ F(S), let Δ(A,B) = (A\B) ∪ (B\A) be the symmetric difference between A and B. Let d(A,B) be the cardinality of Δ(A,B). Is d a metric?
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what have you tried so far? – C Squared Aug 11 '21 at 07:38
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yes but i have problem in third axiome i cant prouve triangle inequality) – malek Aug 11 '21 at 07:40
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Draw a Venn diagram of $3$ partially overlapping discs representing $A,B,C$. There are $7$ pairwise-disjoint regions within $A\cup B\cup C$. Label them $R(1),...,R(7)$. For $1\le n\le 7$ let $P(n)$ be the number of members of $R(n)$. Express $d(A,B),d(B,C),$ and $d(A,C)$ in terms of $P(1),...,P(7).$ – DanielWainfleet Aug 11 '21 at 08:31