Cratylus
Junior Member
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- Aug 14, 2020
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From Pinter’s. A book of set theory
Let A be a well-ordered class. If a ∈ A, let a′ designate the immediate successor of a, and let a′′ designate the immediate predecessor of a (if it exists). Prove the following:
(i)a [MATH]\leqslant[/MATH] b iff a‘ [Math]\leqslant[/MATH] b’
Proof v1.(Inclusion one way)
We have a well ordered set A and if a [Math]\leqslant[/MATH] b. Given a‘ is the Immediate successor of a then a’=a+1. So that a’-1 [Math]\leqslant[/MATH] b =>a’ [Math]\leqslant[/MATH] b+1. Now b’=b+1. Hence
a’ [Math]\leqslant[/MATH] b’
My source for the proof came from
brainly.in
Proof v2
We have a well ordered set so by def 4.50
Given a‘ is the immediate successor of a then a<a‘=> [Math]\nexists c\in C[/MATH] s.t a<c<a’
but a<b so that a’<a<b and a [Math]\nexists c,d\in C,D[/MATH] s.t a’<a<b<c<d<b’ thus a’ [Math]\leqslant[/MATH] b’
This version is shaky.
Any help greatly appreciated.
Let A be a well-ordered class. If a ∈ A, let a′ designate the immediate successor of a, and let a′′ designate the immediate predecessor of a (if it exists). Prove the following:
(i)a [MATH]\leqslant[/MATH] b iff a‘ [Math]\leqslant[/MATH] b’
Proof v1.(Inclusion one way)
We have a well ordered set A and if a [Math]\leqslant[/MATH] b. Given a‘ is the Immediate successor of a then a’=a+1. So that a’-1 [Math]\leqslant[/MATH] b =>a’ [Math]\leqslant[/MATH] b+1. Now b’=b+1. Hence
a’ [Math]\leqslant[/MATH] b’
My source for the proof came from
if a is the predecessor of b, then the value of (a-b) and (b-a) are - Brainly.in
if a is the predecessor of b, then the value of (a-b) and (b-a) are - 3372333
brainly.in
Proof v2
We have a well ordered set so by def 4.50
Given a‘ is the immediate successor of a then a<a‘=> [Math]\nexists c\in C[/MATH] s.t a<c<a’
but a<b so that a’<a<b and a [Math]\nexists c,d\in C,D[/MATH] s.t a’<a<b<c<d<b’ thus a’ [Math]\leqslant[/MATH] b’
This version is shaky.
Any help greatly appreciated.
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