Suppes (10.2) Introduction to Logic, “Definition of Relations’, summary

 

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.]

 

 

 

Summary of

 

Patrick Suppes

 

Introduction to Logic

 

Ch. 10. Relations

 

§10.2 Definition of Relations

 

 

Brief summary:

The order of the terms in a relation is often important. So we cannot just think of a predicate taking more than one term as relating members of some set. For, that set needs to have a fixed order. For example, if the relation is love, then it matters who loves whom, as the feeling is not always mutual. We designate such ordered n-tuples with angle brackets: ⟨x, y⟩. The set that can be substituted for the first term is the domain, for the second term, the counterdomain, and the union of both the domain and the counterdomain is the field.

 

 

Summary

 

In our everyday language, we often talk about relations that hold between a pair of things or that hold among many things.

Thus we say that Elizabeth II stands in the relation of mother to Prince Charles – by which we mean that Elizabeth II is the mother of Prince Charles. Or we may say that the relation of coincidence holds among three lines-by which we mean that they all intersect in one point.

(Suppes 210)

If we used the letter ‘M’ to represent the relation of one being the mother of another, then we might write for the relation ‘Elizabeth II is the mother of Prince Charles’ as

(Elizabeth II) M (Prince Charles)

(Suppes 210)

Or if A means ‘... is the ancestor of ...’:

we may wish to speak of the relation A such that, for every z and y, xAy if and only if z is an ancestor of y.

(Suppes 210)

[When the predicate is for a relation that holds between two terms, it makes sense to write it as standing between them, as shown above. But what if we have three or more terms? Would it be efficient to re-write the letter so that it stands between each? No, instead we can write that letter first and then write all the letters to its right, inside parentheses. This holds when the relation itself is complicatedly made of different sorts of subrelations.]

In dealing with a relation that holds among three or more things, it is convenient to put the letter standing for the relation, and then the names, in proper order, of the things among which it holds. Thus we may speak of the relation P such that, for every x, y, and z, P(x, y, z) if and only if x and y are the parents of z. Similarly, we may speak of the relation B such that, for every x, y, z, and w, B(x, y, z, w) if and only if x owes y dollars to z for w; so that, for example:

B(John, 5, Henry, shoes)

means that John owes Henry five dollars for shoes.

(Suppes 210)

Suppes explains that we will want to speak of relations between things even if we cannot intuitively grasp how that relation might hold between such things.

Thus the set {⟨Aristotle, Λ⟩, ⟨7, Julius Caesar⟩,}

is a relation, although no one would claim it has any intuitive significance.

(Suppes 211)

[Since the relation often involves a certain order to things being related, we define the things under that relation as being a tuple. For example, one person may love a second without that second loving the first. So we need to distinguish the lover from the loved by means of the order of the members in the tuple.]

Our general definition is then:

(I) A binary relation is a set of ordered couples.

According to this definition the relation of loving is the set of ordered couples ⟨x, y⟩ such that x loves y. The relation of being less than is the set of all ordered couples ⟨x, y⟩ of numbers such that, for some positive number z,

x + z = y.

The obvious extension of (I) is that a relation which holds among three things is a set of ordered triples, and a relation which holds among n things is a set of ordered n-tuples.

A relation is called ‘n-ary’ if its members are n-tuples. For the special cases n = 2 and n = 3 we use special names, speaking of ‘binary’ and ‘ternary’ relations.

(Suppes 211)

[The next idea seems to be that since a set of objects take a predicate, we can render this situation also by saying that the set of ordered n-tuples are included as members of the predicate.]

Since a relation is a set of ordered n-tuples, we can also use the “∈” notation to indicate that certain things stand in a given relation. Thus we can write:

⟨John, Mary⟩ ∈ L, instead of:

John L Mary

to indicate that John loves Mary. Similarly we can write:

⟨George, Mary, Elizabeth⟩ ∈ P,

instead of :

P(George, Mary, Elizabeth)

to indicate, let us say, that George and Mary are the parents of Elizabeth.

(Suppes 211)

[I might be missing the next point, but it seems to be the following. An ordered n-tuple by itself is just a set of objects whose order plays a determining factor (while in normal sets it does not). However, whenever we place some ordered n-tuple into a set, it then takes on the status or structure of a relation. We need not specify what that relation is. It is merely assumed that some relation is to be thought of as relating the terms or predicating them.]

It is necessary to remember that an ordered couple is not a relation, but the set consisting of the ordered couple is. For instance,

⟨Thomas Aquinas, 4⟩ is not a relation;

{⟨Thomas Aquinas, 4⟩} is a relation;

{{⟨Thomas Aquinas, 4⟩}} is not a relation.

The last example of the three is not a relation because the only member of the set is itself a set, which is not an ordered couple.

(Suppes 211)

Suppes will now mention some terminology that can prove useful when dealing with binary relations.

[The three concepts deal with the sets of objects that can be substituted for the parts of the binary relation. The set that can be substituted for the first of the two parts, as in the x of ⟨x, y⟩ is called the domain of the relation. The set for the second part (for the y) is the counterdomain or the converse domain. And finally, the field is the combination (the union) of both the sets for the first and second terms.]

If R is a binary relation, then the domain of R – in symbols: D(R) – is the set of all things x such that, for some y, ⟨x, y⟩ ∈ R. Thus if M is the | relation which consists of all couples ⟨x, y⟩ such that x is the mother of y, then the domain of M is the set of all women who are not childless. If

R₁ = {⟨Λ, Plato⟩, ⟨Jane Austen, 101⟩, ⟨the youngest bride in Tibet, Richelieu⟩},

then

D(R₁) = {Λ, Jane Austen, the youngest bride in Tibet}.

The counterdomain (or converse domain) of a binary relation R (in symbols : C(R)) is the set of all things y such that, for some x, ⟨x, y⟩ ∈ R. The counterdomain of the relation M considered just above is the set of all people – since everyone has a mother. If B is the relation which consists of all couples ⟨x, y⟩ such that x is the brother of y, then the domain of B is the set of all men who have at least one brother or sister, and the counterdomain is the set of all people who have at least one brother. We have for the relation R₁ defined above:

C(R₁) = {Plato, 101, Richelieu}.

The field of a binary relation R (in symbols: F(R)) is the union of its domain and its counterdomain. Thus z belongs to the field of a binary relation R if and only if either ⟨x, z⟩ ∈ R for some x or ⟨z, y⟩ ∈ R for some y. The field of the relation B considered just above is the set of all people who belong to families containing at least two children, at least one of which is male. As another example,

F(R₁) = {Λ, Jane Austen, the youngest bride in Tibet, Plato, 101, Richelieu}.

(Suppes 211-212)

 

 

 

From:

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

   

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