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Directory:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
Revision as of 20:52, 11 January 2009
, 20:52, 11 January 2009→1.3.10.3. Propositions and Sentences: markup
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The ''union'' of <math>X\!</math> and <math>Y,\!</math> for two sets <math>X, Y \subseteq U,</math> is denoted by <math>^{\backprime\backprime} \, X \cup Y \, ^{\prime\prime},</math> and defined as the set of elements in <math>U\!</math> that belong to at least one of <math>X\!</math> or <math>Y.\!</math>
The "union" of X and Y, for two sets X, Y ? U, is denoted by "X ? Y" and defined as the set of elements in U that belong to at least one of X or Y.
X ? Y = {u ? U : u ? X or u ? Y }.
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
X \cup Y
& = &
\{ \, u \in U : u \in X\ \operatorname{or}\ u \in Y \, \}.
\\
\end{array}</math>
|}
The "symmetric difference" of X and Y, for two sets X, Y ? U, written "X ? Y", is the set of elements in U that belong to just one of X or Y.
The ''symmetric difference'' of <math>X\!</math> and <math>Y,\!</math> for two sets <math>X, Y \subseteq U,</math> written <math>^{\backprime\backprime} \, X + Y \, ^{\prime\prime},</math> is the set of elements in <math>U\!</math> that belong to just one of <math>X\!</math> or <math>Y.\!</math>
X ? Y = {u ? U : u ? X?Y or u ? Y?X }.
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
X + Y
& = &
\{ \, u \in U : u \in X-Y\ \operatorname{or}\ u \in Y-X \, \}.
\\
\end{array}</math>
|}
<pre>
The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a "sentence" is, they all rely on the reader's native intuition of what a "set" is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance, that just about everybody has of the logical connectives "not", "and", "or", as these are expressed in natural language terms.
The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, but they have to be regarded as almost purely informal in character, at least, at this stage of the game. In particular, these definitions all invoke the undefined notion of what a "sentence" is, they all rely on the reader's native intuition of what a "set" is, and they all derive their coherence and their meaning from the common understanding, but the equally casual use and unreflective acquaintance, that just about everybody has of the logical connectives "not", "and", "or", as these are expressed in natural language terms.