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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 17:44, 24 February 2013
, 17:44, 24 February 2013→6.37. Propositional Types: markup
'''Definition.''' Given a cartesian product <math>X \times Y,\!</math> an ordered pair <math>(x, y) \in X \times Y,\!</math> has the type <math>S \cdot T,\!</math> written <math>(x, y) : S \cdot T,\!</math> if and only if <math>x \in S \subseteq X\!</math> and <math>y \in T \subseteq Y.\!</math> Notice that an ordered pair may have many types.
'''Definition.''' Given a cartesian product <math>X \times Y,\!</math> an ordered pair <math>(x, y) \in X \times Y,\!</math> has the type <math>S \cdot T,\!</math> written <math>(x, y) : S \cdot T,\!</math> if and only if <math>x \in S \subseteq X\!</math> and <math>y \in T \subseteq Y.\!</math> Notice that an ordered pair may have many types.
'''Definition.''' A relation <math>L \subseteq X \times Y\!</math> has type <math>S \cdot T,\!</math> written <math>L : S \cdot T,\!</math> if and only if every <math>(x, y) \in L\!</math> has type <math>S \cdot T,\!</math> that is, if and only if <math>L \subseteq S \times T\!</math> for some <math>S \subseteq X\!</math> and <math>T \subseteq Y.\!</math>
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Notation. "Barred parentheses", like "(" and ")", will be used in pairs to indicate the negations of propositions and the complements of sets. When an n place relation R is initially given relative to the domains X1, ... , Xn and a set S is being mentioned as a subset of one of them, say S c Xi, then the "relevant complement" of S in such a context is the one taken relative to Xi, that is:
Notation. "Barred parentheses", like "(" and ")", will be used in pairs to indicate the negations of propositions and the complements of sets. When an n place relation R is initially given relative to the domains X1, ... , Xn and a set S is being mentioned as a subset of one of them, say S c Xi, then the "relevant complement" of S in such a context is the one taken relative to Xi, that is: