5E. Using predicates to specify subsets

5. Sets, Subsets, and Predicates



69



14) { b ∈ B | 2b ∈ A } =

15) { a ∈ A | a2 ∈ B } =

16) { a ∈ A | a2 < 0 } =

NOTATION 5.29. When talking about sets or using predicates, we usually assume that a

“universe of discourse” U has been agreed on. This means that all the elements of all of the

sets under discussion are assumed to be members of U . Then

{ x | P (x) }

can be used as an abbreviation for { x ∈ U | P (x) }.

The universe of discourse is sometimes assumed to be understood from the context, but it

is an important concept, and it is best to specify it so that there is no room for confusion. For

example, if we say “Everyone is happy,” who is included in this everyone? We usually do not

mean everyone now alive on the Earth. We certainly do not mean everyone who was ever alive

or who will ever live. We mean something more modest: perhaps we mean everyone in the

building, or everyone in the class, or maybe we mean everyone in the room.

Specifying a universe of discourse eliminates this ambiguity. The U is the set of things that

we are talking about. So if we want to talk about people in Lethbridge, we define U to be the

set of all people in Lethbridge. We write this at the beginning of our symbolization key, like

this:

U : the set of all people in Lethbridge

Everything that follows ranges over the universe of discourse. Given this U , “everyone” means

“everyone in Lethbridge” and “someone” means “someone in Lethbridge.”

Each constant names some member of U , so, if U is the set of people in Lethbridge, then

constants Donald, Gregor, and Marybeth can only be used if these three people are all in

Lethbridge. If we want to talk about people in places besides Lethbridge, then we need to

specify a different universe of discourse.

EXAMPLE 5.30. If



U is the set of all Canadian provinces, then

{ x | the English name of x has three syllables }

= {Alberta, New Brunswick, Newfoundland}.



Remark 5.31. There is a very close relationship between sets and unary predicates. In general:

• From any unary predicate P (x), we can define the set

{x | P (x)}.

• Conversely, from any set A, we can define a unary predicate P (x) to be “x is a member

of A.”

Because of this, sets are more-or-less interchangeable with unary predicates. For example, the

predicate “x is a dog” can be symbolized in two quite different ways:

• Our symbolization key could state that D(x) means “x is a dog.”

• Alternatively, our symbolization key could let D be the set of all dogs. Then “x is a

dog” would be translated as “x ∈ D.”

In most of mathematics and computer science we make use of sets, rather than unary predicates.

We will see that this makes it simpler to translate statements from English into First-Order

Logic when quantifiers are involved.



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5. Sets, Subsets, and Predicates



SUMMARY:

• Important definitions:

◦ set

◦ element, member

◦ ordered pair

◦ subset, proper subset

◦ predicate

• A set is unordered and without repetition.

• ∅ and A are subsets of A.

• A = B if and only if we have both A ⊂ B and B ⊂ A.

• For our purposes, predicates usually have only one or two variables.

• If a predicate has two variables, the order of the variables is important.

• Notation:

◦{



}



◦ ∈, ∈

/

◦ ∅ (empty set)

◦ A ⊂ B, A ⊂ B, A ⊃ B

/

◦ #A

◦ P (x), x Q y



(predicates)



◦ { a ∈ A | P (a) }

◦ U (universe of discourse)

◦ (x, y) (ordered pair)

◦ N, N+ , Z, Q, R



Chapter 6



Operations on Sets

While I am interested both in economics and in philosophy, the union of my

interests in the two fields far exceeds their intersection.

Amartya Sen (b. 1933), Nobel prize-winning economist

Autobiography on nobelprize.org



There are several important ways that a new set can be made from sets that you already have.

Any method of doing this is called a set operation.

6A. Union and intersection

Two of the most basic operations are union and intersection. Let us first discuss them in

informal terms. Suppose:

• Alice and Bob are going to have a party, and need to decide who should be invited,

• Alice made a list of all the people that she would like to invite, and

• Bob made a list of all the people that he would like to invite.

Here are two of the many possible decisions they could make.

1) One solution would be to invite everyone that is on either of the lists. That is, they

could begin their invitation list by writing down all of the names on Alice’s list, and

then add all of the names from Bob’s list (or, more precisely, the names from Bob’s

list that are not already included in Alice’s list). This is the union of the lists.

2) A much more conservative solution would be to invite only the people that appear on

both of the lists. That is, they could go through Alice’s list, and cross off everyone

that does not appear on Bob’s list. (They would get the same result by going through

Bob’s list, and crossing off everyone that does not appear on Alice’s list.) This is the

intersection of the lists.

DEFINITION 6.1. Suppose A and B are sets.

1) The union of A and B is the set

A ∪ B = { x | x ∈ A or x ∈ B }.

2) The intersection of A and B is the set

A ∩ B = { x | x ∈ A and x ∈ B }.

Remark 6.2. By drawing the sets A and B as overlapping circles, the union and intersection

can be represented as follows:

71



72



6. Operations on Sets



A



B



A



A ∪ B is shaded



B



A ∩ B is shaded



Pictures like these are called Venn diagrams.

Remark 6.3.

1) In ordinary English, the word “intersection” refers to where two things meet. For

example, the intersection of two streets is where the two streets come together. We

can think of this area as being part of both streets, so this is consistent with the way

the term is used in mathematics.

2) In ordinary English, the word “union” refers to joining things together. For example,

a marriage is the union of two people — it joins the two people into a single married

couple. This is consistent with the way the term is used in mathematics — we could

form the union of Alice’s list and Bob’s list by gluing Bob’s list to the end of Alice’s

list.

EXAMPLE 6.4.

1) {1, 3, 5, 7, 9} ∪ {1, 4, 7, 10} = {1, 3, 4, 5, 7, 9, 10}

2) {1, 3, 5, 7, 9} ∩ {1, 4, 7, 10} = {1, 7}

EXERCISES 6.5. Specify each set by listing its elements.

1) {1, 2, 3, 4} ∪ {3, 4, 5, 6, 7} =

2) {1, 2, 3, 4} ∩ {3, 4, 5, 6, 7} =

3) {p, r, o, n, g} ∩ {h, o, r, n, s} =

4) {p, r, o, n, g} ∪ {h, o, r, n, s} =

5) {1, 3, 5} ∪ {2, 3, 4} ∩ {2, 4, 6} =

6) {1, 3, 5} ∩ {2, 3, 4} ∪ {2, 4, 6} =

Remark 6.6.

1) It is not difficult to see that ∪ and ∩ are commutative. That is, for all sets A and B,

we have

A∪B =B∪A



and



A ∩ B = B ∩ A.



2) It is not difficult to see that ∪ and ∩ are associative. That is, for all sets A, B, and C,

we have

(A ∪ B) ∪ C = A ∪ (B ∪ C)



and



(A ∩ B) ∩ C = A ∩ (B ∩ C).



So there is no need for parenthesis when writing A ∪ B ∪ C or A ∩ B ∩ C.

EXAMPLE 6.7. A Venn diagram can include more than two sets. For example, here are Venn

diagrams of A ∩ B ∩ C and A ∩ (B ∪ C).



6. Operations on Sets

A



B



73

A



C



B



A ∩ B ∩ C is shaded



C



A ∩ (B ∪ C) is shaded



EXERCISE 6.8. Draw Venn diagrams of the indicated sets.

1) A ∪ B ∪ C

2) A ∪ (B ∩ C)

3) (A ∪ B) ∩ C

4) (A ∩ C) ∪ (B ∩ C)

6B. Set difference and complement

The “set difference” is another fundamental operation. (The “complement” is an important

special case.)

EXAMPLE 6.9. If there is a list of people that Alice would like to invite to the party, and also

a list of people that Bob refuses to allow to come to the party (the “veto list”), then it would

be reasonable to invite the people that are on Alice’s list, but not on the veto list. That is,

they could start with Alice’s list, and remove all of the names that are on the veto list. This is

the [ set!difference]set difference of Alice’s list and the veto list.

DEFINITION 6.10. Suppose A and B are sets.

1) The set difference of A and B is the set

A



B = { x ∈ A | x ∈ B } = { x | (x ∈ A) & (x ∈ B) }.

/

/



(Some authors denote this A−B, but that can cause confusion with the usual arithmetic

operation of subtraction.)

2) The complement of B is the set

B=U



B = { x | x ∈ B },

/



where U is the universal set, as usual. (As an alternative, the complement is sometimes

denoted B c , instead of B.)

Remark 6.11. Here are Venn diagrams.

B



A



A



B is shaded



A



A is shaded



B



B



A



B



A



A is shaded



B



B is shaded



74



6. Operations on Sets



EXAMPLE 6.12. Suppose U = PEOPLE is the set of all people.

1) CHILDREN = ADULTS, because adults are the people who are not children.

2) FEMALES



CHILDREN is the set of all adult women.



EXERCISES 6.13. Assume U = {1, 2, 3, . . . , 10}.

Specify each set by listing its elements.

1) {1, 3, 5, 7, 9} {4, 5, 6, 7} =

2) {4, 5, 6, 7}



{1, 3, 5, 7, 9} =



3) {1, 3, 5, 7, 9} =

4) {4, 5, 6, 7} =

EXERCISE 6.14. Draw a Venn diagram of each set.

1) A ∪ B

2) A ∩ B

3) (A

4) A



B)

(B



5) (A ∪ B)



(A



C)



C)

C

6C. Cartesian product



The Cartesian product is another important set operation. Before introducing it, let us recall

the notation for an ordered pair.

NOTATION 6.15. For any objects x and y, mathematicians use (x, y) to denote the ordered

pair whose first coordinate is x and whose second coordinate is y. We have

(x1 , y1 ) = (x2 , y2 ) iff x1 = x2 and y1 = y2 .

EXAMPLE 6.16. A special case of the Cartesian product is familiar to all algebra students:

recall that

(6.17)

R2 = { (x, y) | x ∈ R, y ∈ R }

is the set of all ordered pairs of real numbers. This is the “coordinate plane” (or “xy-plane”)

that is used for graphing functions.

The only functions considered in elementary algebra are from R to R, but this course

considers functions from any set A to any set B. Therefore, it is important to generalize the

above example by replacing the two symbols R in the right-hand side of eq. (6.17) with arbitrary

sets A and B:

DEFINITION 6.18. For any sets A and B, we let

A × B = { (a, b) | a ∈ A, b ∈ B }.

This notation means, for all x, that

x ∈ A × B iff ∃a ∈ A, ∃b ∈ B, x = (a, b).

The set A × B is called the Cartesian product of A and B.