begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §2. OPERATIONS ON SETS. "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, and then paragraph 10 (on ordered pairs) of §2.

ORDERED PAIRS

I note that, in the first part of the paragraph on ordered pairs, Rosenlicht introduces the ordered pair informally - in the same way that I did in the spoken reals. In the latter part of the paragraph, a set theoretic definition is given, , where the three-lined equals sign means that what is on the left is equal by definition to what is on the right.

begin quote ita-1-2-10-pg7 "
This definition does precisely what we want: to any two objects   a, b   (distinct or not) it assigns an object   (a,b),   and it does this in such a fashion that

(a,b) = (c,d)   if and only if   a = c   and   b = d.
"end quote

I would like quickly to verify this. Using the definition of set equality, we need to show that

.

For the left hand direction of the double arrow, we use the equalities   a = c   and   b = d   either   to replace   a,   and   b   on the left side of the set equation with c,   and   d   or   to replace   c,   and   d   on the right side of the set equation with a,   and   b. In both cases the result is an equatioin that says a thing is equal to itself.

For the right hand direction of the double arrow, we must show that the assumption of equality between the sets implies the equalities on the right hand side of the double arrow. If the sets are equal, then each element of one must be an element of the other. This shows first that   a = c   after which it follows that   b = d.