Difference between revisions of "The relationship between logical implication and the subset relation"
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(Marked as dire page, changed subset to \subseteq , added forall x in A[x in B] form, outlined proof of equivalence.) |
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==Definition== | ==Definition== | ||
| − | <math>A\ | + | <math>A\subseteq B</math> (and we say "A is a subset of B") if and only if every element of <math>A</math> also belongs to <math>B</math> |
| − | That is: <math>[A\ | + | That is: <math>[A\subseteq B]\iff\forall x[x\in A\implies x\in B]</math><ref>Definition 3.10 (p10) - Introduction to Set Theory, Third Edition (Revised and Expanded) - Karel Hrbacek and Thomas Jech</ref> |
| + | ===Note: 16/1/2017 by [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 17:36, 16 January 2017 (UTC)=== | ||
| + | We may often write: | ||
| + | * {{M|\forall x\in A[x\in B]}} instead. | ||
| + | This is easily seen to be equivalent as if {{M|A}} is [[empty set|empty]] (so there is no {{M|x\in A}} to speak of) the implication is semantically true, and the forall is vacuously true. | ||
| − | == | + | ==References== |
| − | + | <references/> | |
{{Definition|Set Theory}} | {{Definition|Set Theory}} | ||
Latest revision as of 17:36, 16 January 2017
This page is a dire page and is in desperate need of an update.
Definition
[math]A\subseteq B[/math] (and we say "A is a subset of B") if and only if every element of [math]A[/math] also belongs to [math]B[/math]
That is: [math][A\subseteq B]\iff\forall x[x\in A\implies x\in B][/math][1]
Note: 16/1/2017 by Alec (talk) 17:36, 16 January 2017 (UTC)
We may often write:
- [ilmath]\forall x\in A[x\in B][/ilmath] instead.
This is easily seen to be equivalent as if [ilmath]A[/ilmath] is empty (so there is no [ilmath]x\in A[/ilmath] to speak of) the implication is semantically true, and the forall is vacuously true.
References
- ↑ Definition 3.10 (p10) - Introduction to Set Theory, Third Edition (Revised and Expanded) - Karel Hrbacek and Thomas Jech
