## 23 Apr 2016

### Suppes (10.1) Introduction to Logic, “Ordered Couples”, summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]
[Central Entry Directory]

[The following is summary. My commentary is in brackets. Boldface is mine. I apologize in advance for any distracting typos or other errors.]

Summary of

Patrick Suppes

Introduction to Logic

Ch. 10. Relations

§10.1 Ordered Couples

Brief summary:
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)
We can have ordered triples, quadruples, and so on. Generally speaking, we can have any-numbered ordered n-tuples. We define them all on the basis of ordered couples. An ordered triple, for example, would be:
x, y, z⟩ = ⟨⟨x, y⟩, z
and an ordered n-tuple:
x1, x2, ..., xn⟩ = ⟨⟨x1, x2, ..., xn-1⟩, xn
In sets, the repetition of members does not add new members, but in ordered n-tuples it does. So {1,2,2}={1,2}, but ⟨1,2,2⟩≠⟨1,2⟩. S is a finite sequence only if S is an ordered n-tuple, for example: ⟨Socrates, Plato, Democritus, Aristotle⟩. A Cartesian product is all the possible ordered pairs made by taking each element from one set and pairing it with each member of another. So if A={1,2} and B={Gandhi,Nehru} then
A × B = {⟨1, Gandhi⟩, ⟨1, Nehru⟩, ⟨2, Gandhi⟩, ⟨2, Nehru⟩}

Summary

In chapter 9 we worked with the notions of sets and set membership. All the other notions were defined in terms of these two. Thus “the empty set was defined to be the set Λ such that, for all x, x ∉ Λ; the intersection of two sets was de­fined to be the set of all things which belong to both of the given sets; and so on” (Suppes 208). The new notion we discuss in this chapter, the ordered couple, will not be defined in terms of sets and membership (208). [Recall from section 9.2 that normally in a set, the order of the members does not matter. Now instead, with ordered couples the members will have an ordinal value.]
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)

If we have an ordering of three terms, we would have an ordered triple. More generally we can speak of ordered n-tuples, which we may define in terms of ordered couples (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

In general, then, we can define any ordered n-tuple in the following way:
x1, x2, ..., xn⟩ = ⟨⟨x1, x2, ..., xn-1⟩, xn
(Suppes 209)

Suppes then shows how ordered triples are identical “just when their corresponding members are identical” (209). The two we begin with are
x, y, z⟩ = ⟨u, v, w
[Recall from (2) that ⟨x,y,z⟩=⟨⟨x,y⟩,z⟩.] By (2) we have:
⟨⟨x, y⟩, z⟩ = ⟨⟨u, v⟩, w
(Suppes 209)
[Recall from (1) that ⟨x,y⟩=⟨u,v⟩↔(x=u & y=v).]
We then use (1) to get:
(3)      ⟨x, y⟩ = ⟨u, v
(Suppes 209)
[Here ⟨x, y⟩ and ⟨u, v⟩ of the triplet formulation are equivalent to the x and u of ⟨x,y⟩=⟨u,v⟩↔(x=u & y=v).] Then we further apply (1) to get
z = w
We then use (1) on (3) to get:
x = u and y = v
(Suppes 209)
[Now we have found all the equivalences.]
Thus we have shown:
x, y, z⟩ = ⟨u, v, w⟩ ↔ (x = u & y = v & z = w)
(Suppes 209)
[Recall from section 9.2 that when we repeat an element in a set, it does not count as an additional member.]
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)

[Suppes defines finite sequences as ordered series of items.]
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)

[A Cartesian product seems to be all possible ordered pairs made by taking each element from one set and pairing it with each member of the other.]
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 xA and y B. For example, if
A = {1, 2}
B = {Gandhi, Nehru}
then
A × B = {⟨1, Gandhi⟩, ⟨1, Nehru⟩, ⟨2, Gandhi⟩, ⟨2, Nehru⟩}.
(Suppes 209-210)

From:
Suppes, Patrick. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational, 1957.

.