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begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] 3. FUNCTIONS. "end quote
These pages assume that the reader has the book and is reading along.    
topic index     contents
Please read these notes from the beginning to here.

EXERCISE:     Y Y     XY XY

In preparation for the next paragraph, I would like to prove two facts which Rosenlicht uses.

(1) Let   X, Y, Y   be non-empty sets with   Y Y.   Show that    XY XY.

Step
Fact
Reason
1
If   p = (x,y) XY,    then   xX   and   yY.
definitions:
ordered pair
cartesian product of two sets
2
xX   and   yY
Y Y
definition of subsets
3
p = (x,y) XY
as in step 1
4
XY XY
Given the 'if' part of step 1 and the conclusion of step 3, this follows by the definition of subsets.
Note in particular that this step of this proof is an application of the exercise at the bottom of the subset page:
The lower case   x   of the exercise is replaced by the ordered pair   (x,y)   of this proof, and the upper case   X, Y   of the exercise are replaced by the   XY   and   XY   of this proof respectively.

(2) Let   X, X, Y   be non-empty sets with   X X.   Show that    XY XY.   Try this as an exercise. It is quite similar to case (1) above.

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