734: Set Elements Minus Set

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definition of set elements minus set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of power set of set.
• The reader knows a definition of union of set.

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Target Context

• The reader will have a definition of set elements minus set.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$S^{\prime }$: $\in \{\text{ the sets }\}$
$S$: $\in \{\text{ the sets }\}$
$\ast S^{\prime }{\setminus }_{e}S$: $=\{p\in Pow(\cup S^{\prime })|\mathrm{\exists }{p}^{\prime }\in S^{\prime }(p=p^{\prime }\setminus S)\}$, $\in \{\text{ the sets }\}$
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Conditions:
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2: Natural Language Description

For any set, $S^{\prime }$, and any set, $S$, the set, $S^{\prime }{\setminus }_{e}S=\{p\in Pow(\cup S^{\prime })|\mathrm{\exists }{p}^{\prime }\in S^{\prime }(p=p^{\prime }\setminus S)\}$

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3: Note

More informally, $S^{\prime }{\setminus }_{e}S=\{p^{\prime }\setminus S|p^{\prime }\in S^{\prime }\}$: $S$ is subtracted from each element of $S^{\prime }$. Note that that expression may contain duplicate elements: $p^{\prime }\setminus S=p^{″}\setminus S$ is possible for some $p^{\prime }\ne p^{″}\in S^{\prime }$, which does not make the expression invalid (if not so desirable) though, because the definition of set just automatically eliminates any duplications. We have adopted the formal expression because of the undesirability and also in order to employ the subset axiom.

$S^{\prime }{\setminus }_{e}S$ is indeed a set: $Pow(\cup S^{\prime })$ is a set by the union axiom and the power set axiom; $\mathrm{\exists }{p}^{\prime }\in S^{\prime }(p=p^{\prime }\setminus S)$ is a legitimate formula for the subset axiom.

"set elements minus set" is not any prevalent term, but as we have not seen any prevalent term for the concept, we had to concoct one. "elements minus" means the operator that creates a set from the minuend and the subtrahend. Likewise, ${\setminus }_e$ is not any prevalent notation.

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References

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