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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 02:33, 19 April 2012→6.6. Basic Notions of Group Theory: markup
In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its “polymorphic” practices — from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is used equivocally in a statement like <math>L = L^{-1}(1),\!</math> where a sensible reading requires it to denote the relational set <math>L \subseteq \textstyle\prod_i X_i</math> on the first appearance and the propositional function <math>L : \textstyle\prod_i X_i \to \mathbb{B}</math> on the second appearance.
In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its “polymorphic” practices — from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is used equivocally in a statement like <math>L = L^{-1}(1),\!</math> where a sensible reading requires it to denote the relational set <math>L \subseteq \textstyle\prod_i X_i</math> on the first appearance and the propositional function <math>L : \textstyle\prod_i X_i \to \mathbb{B}</math> on the second appearance.
A '''triadic relation''' is a relation on an ordered triple of nonempty sets. Thus, <math>L\!</math> is a triadic relation on <math>(X, Y, Z)\!</math> if and only if <math>L \subseteq X \times Y \times Z.\!</math> Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation <math>L \subseteq X \times Y \times Z</math> to refer to a logical predicate or a propositional function, of the type <math>X \times Y \times Z \to \mathbb{B},</math> or any one of the derived binary operations, of the types <math>X \times Y \to \operatorname{Pow}(Z),</math> <math>X \times Z \to \operatorname{Pow}(Y),</math> <math>Y \times Z \to \operatorname{Pow}(X).</math>
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A "binary operation" or "law of composition" (LOC) on a nonempty set X is a triadic relation * c XxXxX that is also a function * : XxX >X. The notation "x*y" is used to indicate the functional value *(x, y) C X, which is also referred to as the "product" of x and y under *.
A "binary operation" or "law of composition" (LOC) on a nonempty set X is a triadic relation * c XxXxX that is also a function * : XxX >X. The notation "x*y" is used to indicate the functional value *(x, y) C X, which is also referred to as the "product" of x and y under *.