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Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 21:18, 16 November 2012
, 21:18, 16 November 2012→Relations In General
We have already looked at 2-adic relations that separately exemplify each of these regularities. We also introduced a few bits of additional terminology and special-purpose notations for working with tubular relations.
We have already looked at 2-adic relations that separately exemplify each of these regularities. We also introduced a few bits of additional terminology and special-purpose notations for working with tubular relations.
If <math>L\!</math> is tubular at <math>X,\!</math> then <math>L\!</math> is known as a ''partial function'' or a ''prefunction'' from <math>X\!</math> to <math>Y,\!</math> indicated by writing <math>L : X \rightharpoonup Y.\!</math> We have the following definitions and notations.
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We arrive by way of this winding stair at the special stamps of 2-adic relations <math>L \subseteq X \times Y\!</math> that are variously described as ''1-regular'', ''total and tubular'', or ''total prefunctions'' on specified domains, either <math>X\!</math> or <math>Y\!</math> or both, and that are more often celebrated as ''functions'' on those domains.
We arrive by way of this winding stair at the special stamps of 2-adic relations <math>L \subseteq X \times Y\!</math> that are variously described as ''1-regular'', ''total and tubular'', or ''total prefunctions'' on specified domains, either <math>X\!</math> or <math>Y\!</math> or both, and that are more often celebrated as ''functions'' on those domains.
If <math>L\!</math> is a prefunction <math>L : X \rightharpoonup Y\!</math> that happens to be total at <math>X,\!</math> then <math>L\!</math> is known as a ''function'' from <math>X\!</math> to <math>Y,\!</math> typically indicated as <math>L : X \to Y.\!</math>
If <math>L\!</math> is a prefunction <math>L : X \rightharpoonup Y\!</math> that happens to be total at <math>X,\!</math> then <math>L\!</math> is known as a ''function'' from <math>X\!</math> to <math>Y,\!</math> indicated by writing <math>L : X \to Y.\!</math> To say that a relation <math>L \subseteq X \times Y\!</math> is ''totally tubular'' at <math>X\!</math> is to say that <math>L\!</math> is 1-regular at <math>X.\!</math> Thus, we may formalize the following definitions.
To say that a relation <math>L \subseteq X \times Y\!</math> is ''totally tubular'' at <math>X\!</math> is to say that <math>L\!</math> is 1-regular at <math>X.\!</math> Thus, we may formalize the following definitions:
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{| align="center" cellspacing="8" width="90%"