Ordered pairs

Difference (from prior major revision) (minor diff, author diff)

Removed: 1,3d0

We wish to demonstrate how ordered pairs may be defined in terms of ordinary sets, without introducing new axioms. We begin with a LEMMA: For any objects x, y, z if {x, y} = {x, z} then y = z Proof: y is a member of {x, y}, so y is a member of {x, z}. Therefore either y = z, and we are done, or else y = x. In this case {x, z} = {x, y} = {x} and so z = x = y.

Removed: 5,10d1

We now make the DEFINITION: The ordered pair (x, y) is (x, y) = { {x}, {x, y} } and prove the fundamental THEOREM: For every a, b, x, y, (a, b) = (x, y) if and only if x = a and y = b Proof: It is obvious that if x = a and y = b, then (x, y) = (a, b). To prove the converse, assume that { {a}, {a, b} } = { {x}, {x, y} }. Then {a} is an element of { {x}, {x, y} } so either {a} = {x} or {a} = {x, y}. In the first case x = a and in the second x = y = a. In either case {a} = {x}, so that { {a}, {a, b} } = { {a}, {a, y} }. By our lemma, this implies that {a, b} = {a, y} and so, by the lemma again, b = y, as required.

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