Difference between revisions of "Ordered pair"
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| + | ==Kuratowski definition== | ||
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An ordered pair <math>(a,b)=\{\{a\},\{a,b\}\}</math>, this way <math>(a,b)\ne(b,a)</math>. | An ordered pair <math>(a,b)=\{\{a\},\{a,b\}\}</math>, this way <math>(a,b)\ne(b,a)</math>. | ||
Ordered pairs are vital in the study of [[Relation|relations]] which leads to [[Function|functions]] | Ordered pairs are vital in the study of [[Relation|relations]] which leads to [[Function|functions]] | ||
| − | ==Proof of existence== | + | ===Proof of existence=== |
It is easy to prove ordered pairs exist<br/> | It is easy to prove ordered pairs exist<br/> | ||
Suppose we are given <math>a,b</math> (so we can be sure they exist). | Suppose we are given <math>a,b</math> (so we can be sure they exist). | ||
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The axioms may be found [[Set theory axioms|here]] | The axioms may be found [[Set theory axioms|here]] | ||
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| + | ==Proof of uniqueness== | ||
| + | Before we may write <math>(a,b)</math> we must make sure this is not ambiguous. | ||
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| + | {{Begin Theorem}} | ||
| + | Proof that <math>(a,b)=(a',b')\iff[a=a'\wedge b=b']</math> | ||
| + | {{Begin Proof}} | ||
| + | <math>\impliedby</math> | ||
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| + | Clearly if <math>a=a'</math> and <math>b=b'</math> then <math>(a,b)=\{\{a\},\{a,b\}\}=\{\{a'\},\{a',b'\}\}=(a',b')</math> and we're done. | ||
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| + | <math>\implies</math> | ||
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| + | Assume <math>(a,b)=\{\{a\},\{a,b\}\}=\{\{a'\},\{a',b'\}\}=(a',b')</math>. | ||
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| + | If <math>a\ne b</math> then we must have <math>\{a\}=\{a'\}</math> and <math>\{a,b\}=\{a',b'\}</math> (as clearly <math>\{a\}=\{a',b'\}</math> is false, there are either 2 or 1 elements not contained in <math>\{a\}</math> that are in <math>\{a',b'\}</math> - namely <math>a'</math> and <math>b'</math>) | ||
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| + | Clearly <math>a=a'</math>, then <math>\{a,b\}=\{a',b'\}\implies b=b'</math>. | ||
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| + | If <math>a=b</math> then <math>(a,a)=\{\{a\},\{a,a\}\}=\{\{a\}\}</math>, we know <math>\{\{a\}\}=\{\{a'\},\{a',b'\}\}</math> so again using the [[Set theory axioms]] (namely Extensionality) we see <math>a=a'=b'</math> so <math>a=a'</math> and <math>b=b'</math> holds here too. This completes the proof. | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
{{Definition|Set Theory}} | {{Definition|Set Theory}} | ||
| − | {{Theorem|Set Theory}} | + | {{Theorem Of|Set Theory}} |
Latest revision as of 07:22, 27 April 2015
Kuratowski definition
An ordered pair (a,b)={{a},{a,b}}, this way (a,b)≠(b,a).
Ordered pairs are vital in the study of relations which leads to functions
Proof of existence
It is easy to prove ordered pairs exist
Suppose we are given a,b (so we can be sure they exist).
By the axiom of a pair we may create {a,b} and {a,a}={a}, then we simply have a pair of these, thus {{a},{a,b}} exists.
The axioms may be found here
Proof of uniqueness
Before we may write (a,b) we must make sure this is not ambiguous.
[Expand]
Proof that (a,b)=(a′,b′)⟺[a=a′∧b=b′]
