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MathHisSci (talk | contribs) →Definition: Clarify third condition |
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# For every <math>X, Y \subseteq \Omega</math> with <math> X \subseteq Y</math> and every <math>x \in \Omega \setminus Y</math> we have that <math>f(X\cup \{x\})-f(X)\geq f(Y\cup \{x\})-f(Y)</math>.
# For every <math>S, T \subseteq \Omega</math> we have that <math>f(S)+f(T)\geq f(S\cup T)+f(S\cap T)</math>.
# For every <math>X\subseteq \Omega</math> and <math>x_1,x_2\in \Omega\backslash X</math> such that <math>x_1\neq x_2</math> we have that <math>f(X\cup \{x_1\})+f(X\cup \{x_2\})\geq f(X\cup \{x_1,x_2\})+f(X)</math>.
A nonnegative submodular function is also a [[Subadditive set function|subadditive]] function, but a subadditive function need not be submodular.
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