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, 13:31, 15 April 2013→6.37. Propositional Types: markup
'''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>
'''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>
'''Notation.''' Parentheses in the courier or teletype font, <math>\texttt{( ... )},\!</math> are used to indicate the negations of propositions and the complements of sets. When a <math>k\!</math>-place relation <math>L\!</math> is initially given relative to the domains <math>X_1, \ldots, X_k\!</math> and a set <math>S\!</math> is mentioned as a subset of one of them, say <math>S \subseteq X_j,\!</math> then the ''relevant complement'' of <math>S\!</math> in such a context is the one taken relative to <math>X_j.\!</math> Thus we have the following equivalents.
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:
(S) = S = (Xi S).
{| align="center" cellspacing="8" width="90%"
| <math>\texttt{(} S \texttt{)} ~=~ -\!S ~=~ X_i - S\!</math>
|}
<pre>
When there is occasion for ambiguities that are not resolved by context then one must resort to indices on the bars, as "(S)i", or revert to writing out the intended term in full, as "(Xi S)".
When there is occasion for ambiguities that are not resolved by context then one must resort to indices on the bars, as "(S)i", or revert to writing out the intended term in full, as "(Xi S)".
