by Corry Shores
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[The following is summary. My commentary is in brackets. Boldface is mine. I apologize in advance for any distracting typos or other errors.]
An ordered couple is two objects given in a fixed order. We list the items in a series, separated by commas and placed between angle brackets, for example: ⟨x, y⟩. Two ordered couples are identical just when the first member of one is identical with the first member of the other, and the second member of one is identical with the second member of the other:
⟨x, y⟩ = ⟨u, v⟩ ↔ (x = u & y = v)
⟨x, y, z⟩ = ⟨⟨x, y⟩, z⟩
⟨x1, x2, ..., xn⟩ = ⟨⟨x1, x2, ..., xn-1⟩, xn⟩
A × B = {⟨1, Gandhi⟩, ⟨1, Nehru⟩, ⟨2, Gandhi⟩, ⟨2, Nehru⟩}
Intuitively, an ordered couple is simply two objects given in a fixed order. We use pointed brackets to denote ordered couples. Thus ⟨x, y⟩ is the ordered couple whose first member is x and whose second member is y. In §9.2 we defined two sets as identical when they have the same members. The requirement of identity for ordered couples is stricter. Two ordered couples are identical just when the first member of one is identical with the first member of the other, and the second member of one is identical with the second member of the other. In symbols :(1) ⟨x, y⟩ = ⟨u, v⟩ ↔ (x = u & y = v).We have, for example:{1, 2} = {2, 1}but:⟨1, 2⟩ ≠⟨2, 1⟩(208)
An ordered triple, for instance, is an ordered couple whose first member is an ordered couple, that is,(2) ⟨x, y, z⟩ = ⟨⟨x, y⟩, z⟩.
(Suppes 208)
Then on this basis we can define ordered quadruples:
⟨x, y, z, w⟩ = ⟨⟨x, y, z⟩, w⟩
⟨x1, x2, ..., xn⟩ = ⟨⟨x1, x2, ..., xn-1⟩, xn⟩(Suppes 209)
⟨x, y, z⟩ = ⟨u, v, w⟩
⟨⟨x, y⟩, z⟩ = ⟨⟨u, v⟩, w⟩(Suppes 209)
(3) ⟨x, y⟩ = ⟨u, v⟩(Suppes 209)
z = w
x = u and y = v(Suppes 209)
Thus we have shown:⟨x, y, z⟩ = ⟨u, v, w⟩ ↔ (x = u & y = v & z = w)(Suppes 209)
It is important to notice that the repetition of the same element adds nothing in describing sets but it does in the case of ordered triples. For example,{1, 2, 2} = {1, 2},but⟨1, 2, 2⟩ ≠ ⟨1, 2⟩,since ⟨1, 2, 2⟩ = ⟨⟨1, 2⟩, 2⟩ and ⟨1, 2⟩ ≠ 1.(Suppes 209)
The notion of a finite sequence may be defined in terms of ordered n-tuples. S is a finite sequence if and only if there is a positive integer n such that S is an ordered n-tuple. Thus, for example, ⟨Socrates, Plato, Democritus, Aristotle⟩ is a finite sequence of Greek philosophers. In particular, it is an ordered quadruple.(Suppes 209)
It is often useful to consider the set of all ordered couples which can be formed from two sets in a fixed order. The Cartesian (or cross) product of | two sets A and B (in symbols: A × B) is the set of all ordered couples ⟨x, y⟩ such that x ∈ A and y ∈ B. For example, ifA = {1, 2}B = {Gandhi, Nehru}thenA × B = {⟨1, Gandhi⟩, ⟨1, Nehru⟩, ⟨2, Gandhi⟩, ⟨2, Nehru⟩}.(Suppes 209-210)
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