Suppes (9.5) Introduction to Logic, “Operations on Sets”, 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

 

§9.5 Operations on Sets

 

 

Brief summary:
Certain operations can be performed on sets. If we find all the members shared in common between two sets, we are finding their intersection (∩):

(x)(x ∈ A ∩ B ↔ x ∈ A & x ∈ B)

When two intersecting sets share no members in common, that is, when they are mutually exclusive sets, their intersection is the empty set. The set containing all the members in total from two sets is their union (∪):

(x)(x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B)

All the members in set A that are not in set B is the difference (∼) of A and B.

(x)(x ∈ A ∼ B ↔ x ∈ A & x ∉ B)

These operations can be iterated.

 

 

 

 

Summary

 

 

Suppose A and B are sets. Suppes gives this example: let set A be the set of all Americans and set B be the set of all blue-eyed people. [There are many blue-eyed Americans, thus there are many people who are in both sets. We can think of blue-eyed Americans as forming a third set, understood as lying where the other two sets intersect one another.]

If A and B are sets, then by the intersection of A and B (in symbols: A ∩ B) we mean the set of all things which belong both to A and to B. Thus, for every x, x ∈ (A ∩ B) if and only if x ∈ A and x ∈ B; that is, symbolically:

(1)        (x)(x ∈ A ∩ B ↔ x ∈ A & x ∈ B).

If A is the set of all Americans, and B is the set of all blue-eyed people, then A ∩ B is the set of all blue-eyed Americans.
(Suppes 184)
(184)

 

Now suppose that we have two sets: A the set of all men and B the set of all animals that weigh more than ten tons (184). [Of course, there are no humans who weigh more than ten tons, thus there are no members in the intersection of these two sets.]

In this case we notice that A ∩ B is the empty set (despite the fact that A ≠ Λ, and B ≠ Λ, since some whales weigh more than ten tons). When A ∩ B = Λ, we say that A and B are mutually exclusive.
(184)

 

Suppes then notes that this notion of intersection is similar to how it is used in geometry and algebra. It is like the intersection of two circles, where we want to know which points “lie on both circles” (184d). Also, for the intersection symbol, some people use the algebra symbol for multiplication, the dot ∙ and speak of the “product” of two sets (185a).

 

Now suppose that “A is the set of all animals, and B is the set of all plants” (185). [Here I suppose that their intersection would be an empty set, although I can imagine that there are creatures which are classified as both plants and animals, I am not sure. At any rate, our concern here is not their intersection. What if instead we wanted to know what would the set be of all members taken from A in combination with all members of B? In other words, what is the set that results when we unite these two sets?]

If A and B are sets, then by the union of A and B (in symbols: A ∪ B) we mean the set of all things which belong to at least one of the sets A and B. Thus, for every x, x ∈ (A ∪ B) if and only if either x ∈ A or x ∈ B. (Notice that, as explained in Chapter 1, we use the connective ‘or’ in its non-exclusive sense: ‘x ∈ A or x ∈ B’ is false only in case both x ∉ A and x ∉ B.) Symbolically:

(2)        (x)(x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B).

If A is the set of all animals, and B is the set of all plants, then A ∪ B is the set of all living organisms. One often wishes to consider the union of two sets, however, even when they are not mutually exclusive. For instance, if A is the set of all human adults, and B is the set of all people less than 40 years old, then A ∪ B is the set of all human beings.
(185)

 

Now consider set A, the set of all humans, and set B, the set of all female humans. We might then wonder, what members are in A that are not in B? This would be the set of male humans (185). This is the difference of A and B.

If A and B are two sets, then by the difference of A and B (in symbols: A ∼ B) we mean the set of all things which belong to A but not to B. Thus, for every x, x ∈ A ∼ B if and only if x ∈ A and x ∉ B; that is, symbolically:

(3)    (x)(x ∈ A ∼ B ↔ x ∈ A & x ∉ B).

If A is the set of all human beings, and B is the set of all human females, then A ∼ B is the set of all human males. One often wishes to consider the difference of two sets A and B, however, even when B is not a subset of A. For instance, if A is the set of human beings, and B is the set of all female animals, then A ∼ B is still the set of all human males, and B ∼ A is the set of all female animals which belong to a non-human species.
(185)

 

Suppes then says that these operations on sets can be iterated, and to illustrate, he shows a series of operations performed on a given group of sets:

These operations on sets (intersection, union, and difference) can of course be iterated. Thus, suppose, for instance, that

A = {1, 2},
B = {1, 3, 5},
C = {2, 3, 5, 7},
D = {4, 5, 6, 7};

then

A ∪ B = {1, 2} ∪ {1, 3, 5} = {1, 2, 3, 5}

and hence

C ∩ (A ∪ B) = {2, 3, 5, 7} ∩ {1, 2, 3, 5} = {2, 3, 5}

and hence

D ∼ [C ∩ (A ∪ B)] = {4, 5, 6, 7} ∼ {2, 3, 5} = {4, 6, 7}. |

Similarly, since

C ∪ D = {2, 3, 5, 7} ∪ {4, 5, 6, 7} = { 2, 3, 4, 5, 6, 7},

we have:

(A ∪ B) ∩ (C ∪ D) = {1, 2, 3, 5} ∩ {2, 3, 4, 5, 6, 7} = {2, 3, 5}.
(Suppes 185-186)

 

 

 

 

 

 

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

 

 

 

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