Suppes (9.3) Introduction to Logic, “Inclusion”, summary

by Corry Shores

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[The following is summary. My commentary is in brackets. Boldface is mine.]

 

Summary of

 

Patrick Suppes

 

Introduction to Logic

 

Ch. 9 Sets

 

§9.3 Inclusion

 

 

Brief summary:
Set inclusion is different from set membership. To say one set is included in another is to say that all the terms belonging to the first also belong to the second. However, to say that one set is a member of another set is to say that the first set as a singular whole is one of the terms in the other set. We symbolize set inclusion with the ⊆ symbol. And if all the members of one set are included in a second, but not all of the second are included in the first, then we call the first a proper subset of the second, and we symbolize it with ⊂. We can distinguish set identity, set membership, and set inclusion with these examples, respectively: “Elizabeth II = the present Queen of England,” “Elizabeth II ∈ the class of women,” and “The class of women ⊆ the class of human beings”, noting that despite this symbolic distinction, in everyday language the symbols (=, ∈, and ⊆) in these sentences can all be substituted with the same word, “is”. We can also distinguish set identity, membership, and inclusion using the concepts of symmetry and transitivity. (a) Identity ≠ inclusion, because identity is symmetric but inclusion is not. (b) Inclusion ≠ membership, because inclusion is transitive, but membership is not. And (c) identity ≠ membership, because identity is both symmetric and transitive, but membership is neither.

 

 

Summary

 

Suppose that A and B are sets. And suppose further that “every member of A is also a member of B” (181). In this case, we call A  a subset of B, or we say that A is included in B (181). We symbolize “included in” as ⊆. So consider the sentences:

The set of Americans is a subset of the set of men.
The set of Americans is included in the set of men.

We could render them:

The set of Americans ⊆ the set of men.

And we can symbolically formulate this inclusion relation:

A ⊆ B ↔ (x)(x ∈ A → x ∈ B)

[Read, “Set A is a subset of set B if and only if for all x, if x is included in set A, then x is included in set B”. Or put another way, perhaps, “A is a subset of B only if all members of A are members of B”.] [Now suppose that set A has members {x,y}. If we think of set A twice, then we can say that A is a subset of A, because all the members of A (taken in its first instance of our consideration) are members of A (taken in its second instance of our consideration). Suppes says that it is clear that every set is a subset of itself, but I am not sure that my reasoning above is what makes it so clear.] “It is clear that every set is a subset of itself; i.e., for every set A we have A⊆A” (181). [Recall from the prior section 9.2 that set membership is not transitive. That is to say, from A∈B and B∈C, it does not follow that A∈C. For example, 2∈{1,2} and {1,2}∈{{1,2}, {3,4}}. However, 2∉{{1,2},{3,4}}. But now we learn that set inclusion is transitive. So if A⊆B and B⊆C, then A⊆C. I am not exactly sure how to say why there is this difference. We need of course to distinguish membership from inclusion. My guess is that when we say one set is included in another (⊆) we are saying that the members of the first are members of the second. However, when we say that one set is a member of another (∈), we are saying that the set itself, not its members, belongs to the other set. So membership is the belonging of a group, taken as a whole, within another grouping. But inclusion is the belonging of a group, taken in terms of  the plurality of its members, within another grouping. Again, I am just guessing.]

Moreover, the relation of inclusion is transitive; i.e., if A ⊆ B and B ⊆ C, then A ⊆ C (for if every member of A is a member of B, and every member of B is a member of C, then every member of A is a member of C).
(Suppes 181)

[Recall also that membership was not symmetric, so from A∈B it does not follow that B∈A.] “The relation of inclusion is not symmetric, however; thus {1 , 2} ⊆ {1, 2, 3}, but it is not the case that {1, 2, 3} ⊆ {1, 2}” (181).

 

Suppes then notes that we can infer the distinction between identity, membership, and inclusion from “the questions of symmetry and transitivity” (181). [I think the idea is as follows. We will use the properties of symmetry and transitivity to distinguish identity, membership, and inclusion. So] (a) Identity is symmetric, but inclusion is not symmetric. Therefore, identity and inclusion cannot be the same thing. (b) Inclusion is transitive, but membership is not transitive. Therefore, inclusion and membership are not the same thing. (c) Identity is both symmetric and transitive, but membership is neither symmetric and transitive. Therefore, identity and membership are not the same thing. Yet,

In everyday language all three notions are expressed by the one overburdened verb ‘to be’. Thus in everyday language we write:

Elizabeth II is the present Queen of England,
Elizabeth II is a woman,
Women are human beings.

But in the more exact language being developed here:

Elizabeth II = the present Queen of England,
Elizabeth II ∈ the class of women,
The class of women ⊆ the class of human beings.
(182)

 

Yet, if A⊆B it is still possible that A=B. For, It could also be that B⊆A, “so that A and B have exactly the same members, and hence are identical” (182).

 

Now, whenever A⊆B and yet also A≠B, then we say that A is a proper subset of B, and we symbolize this relation with ⊂. So suppose:

{1, 2} ⊆ {1, 2, 3}

We see that since {1,2} is a subset of {1,2,3}, but {1,2,3} is not a subset of {1,2}, then we can write:

{1, 2} ⊂ {1, 2, 3}

Now instead suppose that we have:

{1, 2, 3} ⊆ {1, 2, 3}

That means it is false to say:

{1, 2, 3} ⊂ {1, 2, 3}
(Suppes 182)

 

We can clarify this distinction with the following formulation:

A ⊂ B ↔ A ⊆ B & A ≠ B
(Suppes 182)

[Read, “Set A is a proper subset of set B if and only if set A is a subset of set B and set A does not equal set B”, or perhaps more simply, “A is a proper subset of B only if all of A’s members are included in B, but not all of B’s members are included in A”.]

 

 

 

 

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

 

 

 

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