Difference between revisions of "Test"
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$$f(A)+f(B)\ge f(A\cap B) + f(A\cup B).\eqno{(2)}$$ | $$f(A)+f(B)\ge f(A\cap B) + f(A\cup B).\eqno{(2)}$$ | ||
</wikitex> | </wikitex> | ||
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+ | Property test: [[testproperty::Dummypage]] | ||
{{#widget:YouTube|id=t2PsiJXswiM}} | {{#widget:YouTube|id=t2PsiJXswiM}} |
Revision as of 16:47, 30 November 2011
<wikitex> Let $Q$ be any finite set, and $\mathcal B=2^Q$ be the collection of the subsets of $Q$. Let $f:\mathcal B\rightarrow \mathbb R$ be a function assigning real numbers to the subsets of $Q$ and suppose $f$ satisfies the following conditions:
- (i) $f(A)\ge 0$ for all $A\subseteq Q$, $f(\emptyset)=0$,
- (ii) $f$ is monotone, i.e. if $A\subseteq B\subseteq Q$ then $f(A)\le f(B)$,
- (iii) $f$ is submodular, i.e. if $A$ and $B$ are different subsets of $Q$ then
$$f(A)+f(B)\ge f(A\cap B) + f(A\cup B).\eqno{(2)}$$
</wikitex>
Property test: Dummypage
<wikitex> <math>\frac{1}{\displaystyle1+\frac{1}{\displaystyle 1+\sqrt{5}}}</math> </wikitex>