Subsets, Intersections, and Unions

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Note that every object that is not a set (e.g., every physical object) has no members. Ex-tensionality does not make these objects equal to ∅, because extensionality applies only to sets.

5.1.2 Subsets, Intersections, and Unions

If X and Y are sets than we say that X is a subset of Y if every member of X is a member of Y . We also say in that case that Y is a superset of X. The notation is:

X ⊆ Y, or, equivalently, Y ⊇ X .

Occasionally, we use the term inclusion: we say that X is included in Y , meaning that X is a subset of Y .

As is usual in mathematics, crossing out indicates negation:

X 6⊆ Y means that X is not a subset of Y .

Obviously, X⊆ X, for every set X.

Proper Subsets: If X ⊆ Y and X 6= Y , then X is said to be a proper subset of Y , or properly included in Y , and Y is said to be a proper superset of X.

If X ⊆ Y and Y ⊆ X, then X and Y have the same members and, by extensionality, are the same. Therefore

X = Y iff X ⊆ Y and Y ⊆ X.

It is convenient to “chain” inclusions thus: X ⊆ Y ⊆ Z; it means: X ⊆ Y and Y ⊆ Z. Set inclusion is transitive:

If X ⊆ Y ⊆ Z then X ⊆ Z.

(The proof is trivial: Assume the left hand side. If x∈ X then x ∈ Y , because X ⊆ Y ; hence also x∈ Z, because Y ⊆ Z; therefore every member of X is a member of Z.)

Every set, X, contains as members all members of the empty set (because the empty set has no members). Hence,

∅ ⊆ X, for every set X

Note: The subset relation, ⊆ , should be sharply distinguished from the membership relation, ∈ . Every set is a subset of itself, but not a member of itself. On the other hand, a member of a set need not be a subset of it; the earth is a member of {Earth, Moon}, but it is not a subset of it, because the earth is not a set. Or consider the following:

∅ ⊆ {{∅}} (and the inclusion is proper), but ∅ 6∈ {{∅}}; because the only member of {{∅}} is {∅}, and ∅ 6= {∅}.

{∅} ∈ {{∅}} but {∅} 6⊆ {{∅}}; because {∅} contains ∅ as a member, whereas {{∅}} does not contain ∅ as a member.

Intersections

The intersection of two sets X and Y , denoted X∩ Y , is the set whose members are all the objects that are members both of X and of Y :

For every x, x∈ X ∩ Y iff x∈ X and x ∈ Y . or, equivalently:

X∩ Y = {x : x ∈ X and x ∈ Y }

Examples: The intersection of the set of all natural numbers divisible by 2 and the set of all natural numbers divisible by 3 is the set of all natural numbers divisible both by 2 and by 3. (This is the set of natural numbers divisible by 6.)

The intersection of the set of all even natural numbers and the set of all prime numbers is the set of all numbers that are both even and prime; since the only number that is both even and prime is 2, this is the singleton {2}.

The intersection of the set of all USA citizens and the set of all redheaded people is the set of all redheaded USA citizens.

The intersection of the set of all women and the set of all pre-1992 USA presidents is the set of all women that have been, at some time before 1992, USA presidents. This happens to be the empty set.

Disjoint Sets: Two sets, X, Y , are said to be disjoint if they have no common members;

i.e., if X ∩ Y = ∅.

Unions

The union of the sets X and Y , denoted X∪ Y , is the set whose members are all objects that are either members of X or members of Y (or members of both). That is:

For every x, x∈ X ∪ Y iff x∈ X or x ∈ Y . or, equivalently:

X∪ Y = {x : x ∈ X or x ∈ Y }

Examples: The union of the set of all natural numbers that are divisible by 6 and the set of all natural numbers that are divisible by 4 is the set of all numbers divisible either by 6 or by 4 (or by both, e.g., 12).

The union of the set of all mammals and the set of all humans is the set of all creatures that are either mammals or humans; since every human is a mammal, this union is the set of all mammals.

The union of the set of all people that were, at some time up to t, senators, and the set of all people who were, at some time up to t, congressmen, is the set of people who were at one time or another, up to time t, members of at least one of the legislative houses.

The basic properties of intersections and unions are the following:

(X∩ Y ) ∩ Z = X ∩ (Y ∩ Z) (X∪ Y ) ∪ Z = X ∪ (Y ∪ Z)

X∩ Y = Y ∩ X X∪ Y = Y ∪ X

X∩ X = X X∪ X = X

X∩ ∅ = ∅ X∪ ∅ = X

The equalities of the first row mean that the operations of intersection and union are asso-ciative, those of the second row mean that they are commutative, and those of the third row – that they are idempotent. These properties follow directly from the meanings of ‘and’ and

‘or’. They are so obvious that one would hardly consider proving them formally. Formal, but tedious, proofs can be given. When this is done, one sees that the associativity of intersection reflects the associativity of ‘and’ (i.e., the fact that (A∧ B) ∧ C and A ∧ (B ∧ C) are logically equivalent) and the associativity of union reflects that of ‘or’.

Repeated Intersections and Repeated Unions: Intersections can be applied repeatedly to more than two sets, and the same holds for unions. Since these operations are associative, we can ignore grouping and use expressions such as:

X1∩ X²∩ . . . ∩ Xⁿ X1∪ X²∪ . . . ∪ Xⁿ

And since the operations are commutative, the order of the sets can be changed without affecting the result.

It is easily seen that X1∩ X²∩ . . . ∩ Xⁿ is the set of all objects that are members of all the sets X1, . . . , Xn. Similarly, X1∪ X²∪ . . . ∪ Xⁿ is the set of all objects that are members of at least one of X1, . . . , Xn.

Distributive Laws: These two equalities hold in general:

X∩ (Y ∪ Z) = (X ∩ Y ) ∪ (X ∩ Z) X∪ (Y ∩ Z) = (X ∪ Y ) ∩ (X ∪ Z) The first is the distributive law of intersection over union, the second – of union over inter-section. These laws are direct outcomes of the following two tautologies:

x∈ X and (x ∈ Y or x ∈ Z) iff (x ∈ X and x ∈ Y ) or (x ∈ X and x ∈ Z).

x∈ X or (x ∈ Y and x ∈ Z) iff (x ∈ X or x ∈ Y ) and (x ∈ X or x ∈ Z).

Obviously, each of X and Y includes (as a subset) their intersection X ∩ Y , and is included in their union X∪ Y . Which can be stated thus:

X∩ Y ⊆ X, Y ⊆ X ∪ Y

As is easily seen, the subset relation can be characterized in terms either of unions, or of intersections:

X⊆ Y iff X ∩ Y = X X ⊆ Y iff X ∪ Y = Y

We also have:

If X ⊆ X⁰ and Y ⊆ Y⁰ then X∩ Y ⊆ X⁰∩ Y⁰ and X∪ Y ⊆ X⁰∪ Y⁰.

Every set which is included both in X and in Y is included in their intersection. This follows easily from the definitions. (It is also derivable from the above-given properties: If Z ⊆ X and Z ⊆ Y , then Z = Z ∩ Z ⊆ X ∩ Y .)

Therefore, the intersection of two sets X and Y is (i) included both in X and in Y , and

(ii) includes every set that is included in X and in Y .

We can express this by saying that X∩ Y is the largest set that is included both in X and in Y .

Similarly, the union of X and Y can be characterized as the smallest set that includes both X and Y .

Homework

5.2 Let N ={0, 1, 2, . . . , n, . . .} and let x, y, z, range over N. Let X1 ={0, 1, 5, 7, 10, 13, 18, 19, 20}

X2 ={3, 4, 5, 17, 21, 8, 9, 6, 1}

X3 ={21, 31, 20, 40, 1, 0, 3, 20}

X4 ={2x : x > 3}

X5 ={x : x is divisible by 2 or by 3}

X6 ={x : x is prime}

Write down in the curly-bracket notation (using ‘∅’ for the empty set) the following sets:

1. X1∪ X² 2. X1∩ X³ 3. X3∩ X⁴ 4. X3∪ X⁴ 5. X1∩ X²∩ X³

6. (X1∩ X⁶)∪ (X⁵∩ X²) 7. (X5∩ X⁶)∪ X¹

8. (X6∪ X⁵)∩ (X¹∪ X³) 9. (X4∩ X⁶)∪ X⁵

10. X4∩ (X⁶ ∪ X⁵)

5.3 For any two sets, X, Y , define X− Y by:

X− Y = {x ∈ X : x 6∈ Y }

With the Xi’s as in 5.2, write down in the curly-bracket notation (using ‘∅’ for the empty set) the following sets:

1. X1− X² 2. X2− X¹ 3. X6− X⁵ 4. X4− X⁵

5. (X3− X¹)∩ (X²− X⁴) 6. (X1− X³)− X²

7. X1− (X³ − X²)

8. N− X⁴ 9. X4− N

10. X5− (X⁶∪ X⁴)