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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.

THE       if       AND       only if       PARTS       OF       if and only if

I would like to explain the   if   and   only if   parts of an   if and only if   statement. However, I would like to do this without letters because I do not want the symbols to have an assumed order - as would be the case if we use   A   and   B.   Therefore, I will use     and   .

Statements to be proved will often be in the form   if and only if .   Recall from the first reading of 1 that the words   if and only if   can be replaced by a double arrow which should be read in both directions, and each direction is a material implication.

I read
if and only if
beginning with the first occurrence of the word   if   to get

if     then  
and this is the   if   part.

Note that we read to the end and then go back to the beginning to complete the sentence.
Then I recall that the implication also has to go the other way:

if     then  
and this is the   only if   part.

These two parts must be proved separately because the two implications do not follow from each other.

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