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C, D
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Concepts or Classes
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\(C^{{\mathcal {I}}},D^{{\mathcal {I}}}\subseteq \Theta\)
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R, S
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Relations or Properties
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\(R^{{\mathcal {I}}},S^{{\mathcal {I}}}\subseteq \Theta \times \Theta\)
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\(C\sqsubseteq D\)
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Class subsumption
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\(C^{{\mathcal {I}}}\subseteq D^{{\mathcal {I}}}\)
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\(R\sqsubseteq S\)
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Property subsumption
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\(R^{{\mathcal {I}}}\subseteq S^{{\mathcal {I}}}\)
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\(C\sqcup D\)
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Union of classes
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\((C\sqcup D)^{\mathcal {I}}=C^{{\mathcal {I}}}\cup D^{\mathcal {{\mathcal {I}}}}\)
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\(C \sqcap D\)
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Intersection of classes
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\((C \sqcap D)^{{\mathcal {I}}}=C^{{\mathcal {I}}} \cap D^{{\mathcal {I}}}\)
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\(\lnot C\)
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Complement
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\((\lnot C)^{\mathcal {I}}=\Theta - C^{{\mathcal {I}}}\)
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\(\exists R.C\)
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Existential quantification
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\({(\exists R.C)}^{\mathcal {I}}=\{a \in \Theta | \exists b, \langle a, b \rangle \in R^{\mathcal {I}} \wedge b \in C^{\mathcal {I}}\}\)
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\(\forall R.C\)
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Value restriction
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\({(\forall R.C)}^{\mathcal {I}}=\{a \in \Theta | \forall b, \langle a, b \rangle \in R^{\mathcal {I}} \rightarrow b \in C^{{\mathcal {I}}}\}\)
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\(\top\)
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Top concept or Thing
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\(\top ^{{\mathcal {I}}}=\Theta\)
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\(\bot\)
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Bottom concept or Nothing
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\(\bot ^{{\mathcal {I}}}=\varnothing\)
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