推 FatDaniel :這樣好像只有說到subset的cardinality是finite狀況? 01/16 08:46
推 bineapple :試試反證 01/16 13:43
Prove that every nonempty set of positive integers contains a smallest memeber.
我嚐試做的證明如下:
Proof:
+
Let the set be S⊆Z.
Prove by induction. +
Basis step: |S| = 1 => S = {a}, a ∈ Z. Obviously, a is the smallest memeber.
Inductive step:
Assume |S| = n, there is a smallest memeber b.
Then, when |S| = n + 1, let c be an additional memeber.
(By Axiom 6: Exactly one of the relations x = y, x < y, x > y holds.)
If c ≧ b, then choose b as the smallest member.
If c < b,then choose c as the smallest member.
Then, there is a smallest member in S.
By induction, the statement is true.
看起來好像是對的,但如果S的元素數無窮大,會不會出問題?
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 111.251.161.172