User:IssaRice/Partial order summary table - Revision history

IssaRice at 19:48, 2 October 2021

2021-10-02T19:48:01Z

← Older revision		Revision as of 19:48, 2 October 2021
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	\| Maximum element \|\| <math>y_0 \in Y</math> such that <math>y_0 \geq y</math> for every <math>y \in Y</math> (i.e. an upper bound which happens to be in the set) \|\| Yes		\| Maximum element \|\| <math>y_0 \in Y</math> such that <math>y_0 \geq y</math> for every <math>y \in Y</math> (i.e. an upper bound which happens to be in the set) \|\| Yes
	\|-		\|-
	\| ~~Supremum~~ \|\| \|\| No		\| Least upper bound \|\| an upper bound <math>M</math> such that if <math>K</math> is another upper bound for <math>Y</math>, then <math>M\leq K</math> \|\| No
	\|-		\|-
	\| ~~Least~~ upper bound \|\| \|\| No		\| Supremum \|\| (same as least upper bound, though in some cases like on the real line, the least upper bound is thought of as being a real number and will not exist when a set is not bounded above, whereas the supremum always exists) \|\| No
	\|}		\|}

IssaRice at 19:44, 2 October 2021

2021-10-02T19:44:42Z

← Older revision		Revision as of 19:44, 2 October 2021
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	\| Maximal element \|\| <math>y_0 \in Y</math> such that there exists no <math>y \in Y</math> for which <math>y > y_0</math> \|\| Yes		\| Maximal element \|\| <math>y_0 \in Y</math> such that there exists no <math>y \in Y</math> for which <math>y > y_0</math> \|\| Yes
	\|-		\|-
	\| Maximum element \|\| <math>y_0 \in Y</math> such that <math>y_0 \geq y</math> for every <math>y \in Y</math> \|\| Yes		\| Maximum element \|\| <math>y_0 \in Y</math> such that <math>y_0 \geq y</math> for every <math>y \in Y</math> (i.e. an upper bound which happens to be in the set) \|\| Yes
	\|-		\|-
	\| Supremum \|\| \|\| No		\| Supremum \|\| \|\| No

IssaRice at 19:43, 2 October 2021

2021-10-02T19:43:03Z

← Older revision		Revision as of 19:43, 2 October 2021
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	Let <math>(X, \leq)</math> be a partially ordered set, and let <math>Y \~~subset~~ X</math> be a subset of <math>X</math>.		Let <math>(X, \leq)</math> be a partially ordered set, and let <math>Y \subseteq X</math> be a subset of <math>X</math>.

	{\| class="wikitable"		{\| class="wikitable"

IssaRice at 19:42, 2 October 2021

2021-10-02T19:42:53Z

← Older revision		Revision as of 19:42, 2 October 2021
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	\| Maximal element \|\| <math>y_0 \in Y</math> such that there exists no <math>y \in Y</math> for which <math>y > y_0</math> \|\| Yes		\| Maximal element \|\| <math>y_0 \in Y</math> such that there exists no <math>y \in Y</math> for which <math>y > y_0</math> \|\| Yes
	\|-		\|-
	\| Maximum element \|\| <math>y_0 \in Y</math> such that <math>y_0 \~~leq~~ y</math> for every <math>y \in Y</math> \|\| Yes		\| Maximum element \|\| <math>y_0 \in Y</math> such that <math>y_0 \geq y</math> for every <math>y \in Y</math> \|\| Yes
	\|-		\|-
	\| Supremum \|\| \|\| No		\| Supremum \|\| \|\| No

IssaRice at 19:42, 2 October 2021

2021-10-02T19:42:35Z

← Older revision		Revision as of 19:42, 2 October 2021
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	Let <math>(X, \leq)</math> be a partially ordered set.		Let <math>(X, \leq)</math> be a partially ordered set, and let <math>Y \subset X</math> be a subset of <math>X</math>.

	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
	! Term !! Definition !! Must be in set?		! Term !! Definition !! Must be in set <math>Y</math>?
	\|-		\|-
	\| Upper bound \|\| \|\| No		\| Upper bound \|\| <math>M \in X</math> such that <math>y \leq M</math> for all <math>y \in Y</math> \|\| No
	\|-		\|-
	\| Maximal element \|\| \|\| Yes		\| Maximal element \|\| <math>y_0 \in Y</math> such that there exists no <math>y \in Y</math> for which <math>y > y_0</math> \|\| Yes
	\|-		\|-
	\| Maximum element \|\| \|\| Yes		\| Maximum element \|\| <math>y_0 \in Y</math> such that <math>y_0 \leq y</math> for every <math>y \in Y</math> \|\| Yes
	\|-		\|-
	\| Supremum \|\| \|\| No		\| Supremum \|\| \|\| No

IssaRice at 19:36, 2 October 2021

2021-10-02T19:36:39Z

← Older revision		Revision as of 19:36, 2 October 2021
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	Let <math>(X, \leq)</math> be a partially ordered set.		Let <math>(X, \leq)</math> be a partially ordered set.

	{\| ~~style~~="wikitable"		{\| class="wikitable"
	\|-		\|-
	! Term !! Definition !! Must be in set?		! Term !! Definition !! Must be in set?

IssaRice: Created page with "Let $(X, \leq)$ be a partially ordered set. {| style="wikitable" |- ! Term !! Definition !! Must be in set? |- | Upper bound || || No |- | Maximal element || || Ye..."

2021-10-02T19:36:25Z

Created page with "Let <math>(X, \leq)</math> be a partially ordered set. {| style="wikitable" |- ! Term !! Definition !! Must be in set? |- | Upper bound || || No |- | Maximal element || || Ye..."

New page

Let <math>(X, \leq)</math> be a partially ordered set.

{| style="wikitable"
|-
! Term !! Definition !! Must be in set?
|-
| Upper bound || || No
|-
| Maximal element || || Yes
|-
| Maximum element || || Yes
|-
| Supremum || || No
|-
| Least upper bound || || No
|}

User:IssaRice/Partial order summary table - Revision history

IssaRice at 19:48, 2 October 2021

IssaRice at 19:44, 2 October 2021

IssaRice at 19:43, 2 October 2021

IssaRice at 19:42, 2 October 2021

IssaRice at 19:42, 2 October 2021

IssaRice at 19:36, 2 October 2021

IssaRice: Created page with "Let (X, \leq) be a partially ordered set. {| style="wikitable" |- ! Term !! Definition !! Must be in set? |- | Upper bound || || No |- | Maximal element || || Ye..."

IssaRice: Created page with "Let $(X, \leq)$ be a partially ordered set. {| style="wikitable" |- ! Term !! Definition !! Must be in set? |- | Upper bound || || No |- | Maximal element || || Ye..."