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begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] 4. FINITE AND INFINITE SETS. "end quote
These pages assume that the reader has the book and is reading along.    
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Please read these notes from the beginning to here and then the first 3 sentences of paragraph 4 of 4.

PROPER SUBSETS OF INFINITE SETS

First please recall that the meaning of the material implication     is   ,   where   ~   means   "not"   and where the symbol     means "and".   Now recall that all important substitution exercise that was first encountered on the subset page.   Replace   A   above by   ~B   and replace   B   by   ~A.   We get the following:


~B~A       means       ~(    ~B ~(~A)   ).

But   ~(~A)   is just   A   and the symbol     is commutative, which means we can read the "and" statement either from right to left or from left to right. Consequently we see that   "not B implies not A"   has exactly the same meaning as   "A implies B".   Rosenlicht uses this here:

begin quote ita-1-4-4-pg11 "
It is easy to show that a set   X   is infinite if and only if it may be put into one-one correspondence with a proper subset of itself. To do this, note first that if   X   is finite then any proper subset has a smaller number of elements, whereas two finite sets in one-one correspondence must have the same number of elements. This proves the   "if"   part.
"end quote

The   if part   is the statement   "If   X   can be put into one-one correspondence with a proper subset of itself, then   X   is infinite." To use the result presented above, we negate and switch:   "If   X   is finite, then it can't be put into one-one correspondence with a proper subset of itself."   This is what Rosenlicht proves.

on to reading 4

(for more on the   if   and   only if   parts)