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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

AandBare sets, then by theintersectionofAandB(in symbols:A∩B) we mean the set of all things which belong both toAand toB. Thus, for everyx,x∈ (A∩B) if and only ifx∈Aandx∈B; that is, symbolically:(1) (

x)(x∈A∩B↔x∈A&x∈B).If

Ais the set of all Americans, andBis the set of all blue-eyed people, thenA∩Bis 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∩Bis the empty set (despite the fact thatA ≠Λ, andB≠ Λ, since some whales weigh more than ten tons). WhenA∩B= Λ, we say thatAandBaremutually 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

AandBare sets, then by the union ofAandB(in symbols:A∪B) we mean the set of all things which belong to at least one of the setsAandB. Thus, for everyx,x∈ (A∪ B) if and only if eitherx∈Aorx∈B. (Notice that, as explained in Chapter 1, we use the connective ‘or’ in itsnon-exclusivesense: ‘x∈Aorx∈B’ is false only in case bothx∉Aandx∉B.) Symbolically:(2) (

x)(x∈A∪B↔x∈A∨x∈B).If

Ais the set of all animals, andBis the set of all plants, thenA∪Bis 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, ifAis the set of all human adults, andBis the set of all people less than 40 years old, thenA∪Bis 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

AandBare two sets, then by thedifferenceofAandB(in symbols:A∼B) we mean the set of all things which belong toAbut not toB. Thus, for everyx,x∈A∼Bif and only ifx∈Aandx∉B; that is, symbolically:(3) (

x)(x∈A∼B↔x∈A&x∉B).If

Ais the set of all human beings, andBis the set of all human females, thenA∼Bis the set of all human males. One often wishes to consider the difference of two setsAandB, however, even whenBis not a subset ofA. For instance, ifAis the set of human beings, andBis the set of all female animals, thenA∼Bis still the set of all human males, andB∼Ais 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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