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StatOnTheSide
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Proof of "If S(m)=S(n), then m=n"
Hello all. I have a question regarding the statement
If S(m)=S(n), then m=n, where S(m)=m[itex]\cup[/itex]{m}, the successor of the natural number m.
I have come across two proofs for the above.
1. This one is as simple as observing the fact that
if S(m)=S(n), then [itex]\bigcup[/itex]S(m)=[itex]\bigcup[/itex]S(n) and since m and n are transitive (a set A is transitive if x[itex]\in[/itex]a[itex]\in[/itex]A[itex]\Rightarrow[/itex]x[itex]\in[/itex]A), it implies that m=n (for transitive sets like A, it can be proved that [itex]\bigcup[/itex]S(A)=A).
This cheeky and cute proof is from Enderton's book.
2. This proof is a longer one which is given in Halmos' book and Hrbacek and Jech's book
follows a similar line of thought.
It involves proving two Lemmas. To quote him,
This is basically proving the ideas related to order defined on the set of natural numbers.
I wish to know the reason, if there is any, as to why one proof might be better than the other. Proper perspective always helps in understanding abstract mathematical concepts. Your input is greatly appreciated.
Hello all. I have a question regarding the statement
If S(m)=S(n), then m=n, where S(m)=m[itex]\cup[/itex]{m}, the successor of the natural number m.
I have come across two proofs for the above.
1. This one is as simple as observing the fact that
if S(m)=S(n), then [itex]\bigcup[/itex]S(m)=[itex]\bigcup[/itex]S(n) and since m and n are transitive (a set A is transitive if x[itex]\in[/itex]a[itex]\in[/itex]A[itex]\Rightarrow[/itex]x[itex]\in[/itex]A), it implies that m=n (for transitive sets like A, it can be proved that [itex]\bigcup[/itex]S(A)=A).
This cheeky and cute proof is from Enderton's book.
2. This proof is a longer one which is given in Halmos' book and Hrbacek and Jech's book
follows a similar line of thought.
It involves proving two Lemmas. To quote him,
Naive Set Theory, P. R. Halmos, pp.47
(i) no natural number is a subset of any of its elements, and (ii)
every element of a natural number is a subset of it.
This is basically proving the ideas related to order defined on the set of natural numbers.
I wish to know the reason, if there is any, as to why one proof might be better than the other. Proper perspective always helps in understanding abstract mathematical concepts. Your input is greatly appreciated.