Difference between revisions of "Template:AotD"

Revision as of 22:00, 19 December 2007

Zorn's Lemma

Zorn's Lemma is a set theoretic result which is equivalent to the Axiom of Choice.
Let $A$ be a partially ordered set.
We say that $A$ is inductively ordered if every totally ordered subset $T$ of $A$ has an upper bound, i.e., an element $a \in A$ such that for all $x\in T$ , $x \le a$ . We say that $A$ is strictly inductively ordered if every totally ordered subset $T$ of $A$ has a least upper bound, i.e., an upper bound $a$ so that if $b$ is an upper bound of $T$ , then $a \le b$ .
An element $m \in A$ is maximal if the relation $a \ge m$ implies $a=m$ . (Note that a set may have several maximal... [more]

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-===[[Diophantine equation]]===
+==[[Zorn's Lemma]]==
-{{WotWAlso}}
+'''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to the [[Axiom of Choice]].
-A '''Diophantine equation''' is an multi-variable [[equation]] for which [[integer]] solutions (or sometimes [[natural number]] or [[whole number]] solutions) are to be found.
-Finding the solution or solutions to a Diophantine equation is closely tied to [[modular arithmetic]] and [[number theory]]. Often, when a Diophantine equation has infinitely many solutions, [[parametric form]] is used to express the relation between the variables of the equation.
+Let <math>A</math> be a [[partially ordered set]].
-Diophantine equations are named for the ancient Greek/Alexandrian mathematician Diophantus.
+We say that <math>A</math> is ''inductively ordered'' if every [[totally ordered set | totally ordered]] [[subset]] <math>T</math> of <math>A</math> has an upper bound, i.e., an element <math>a \in A</math> such that for all <math>x\in T</math>, <math>x \le a</math>.  We say that <math>A</math> is ''strictly inductively ordered'' if every totally ordered subset <math>T</math> of <math>A</math> has a [[least upper bound]], i.e., an upper bound <math>a</math> so that if <math>b</math> is an upper bound of <math>T</math>, then <math>a \le b</math>.
-A Diophantine equation in the form <math>ax+by=c</math> is known as a linear combination.  If two [[relatively prime]] integers <math>a</math> and <math>b</math> are written in this form with <math>c=1</math>, the equation will have an infinite number of solutions.  More generally, there will always be an... [[Diophantine equation|[more]]]
+An element <math>m \in A</math> is [[maximal element | maximal]] if the relation <math>a \ge m</math> implies <math>a=m</math>.  (Note that a set may have several maximal... [[Zorn's Lemma|[more]]]
 </blockquote>

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Difference between revisions of "Template:AotD"

Revision as of 22:00, 19 December 2007

Zorn's Lemma