Page 91 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More Operations on Sets 77
occurred inside the power set notation, so we wrote out the definition of power
set first. In statement 5, the power set notation occurred within both sides of
the notation for the intersection of two sets, so we started with the definition of
intersection. Similar considerations led us to use the definition of subset first,
rather than power set, in statement 2.
It is interesting to note that our answers for statements 4 and 5 in
Example 2.3.3 are equivalent. (You are asked to verify this in exercise 10.) As in
Section 1.4, it follows that for any sets A and B, P (A ∩ B) = P (A) ∩ P (B).
You are asked in exercise 11 to show that this equation is not true in general if
we change ∩ to ∪.
Consider once again the family of sets F ={C s | s ∈ S}, where S is the set
of all students and for each student s, C s is the set of all courses that s has taken.
If we wanted to know which courses had been taken by all students, we would
need to find those elements that all the sets in F have in common. The set of all
these common elements is called the intersection of the family F and is written
∩F. Similarly, the union of the family F, written ∪F, is the set resulting from
throwing all the elements of all the sets in F together into one set. In this case,
∪F would be the set of all courses that had been taken by any student.

Example 2.3.4. Let F ={{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}}. Find ∩F and
∪F.

Solution

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

Although these examples may make it clear what we mean by ∩F and ∪F,
we still have not given careful definitions for these sets. In general, if F is any
family of sets, then we want ∩F to contain the elements that all the sets in F
have in common. Thus, to be an element of ∩F, an object will have to be an
element of every set in F. On the other hand, anything that is an element of
any of the sets in F should be in ∪F,sotobein ∪F an object only needs to be
an element of at least one set in F. Thus, we are led to the following general
definitions.

Definition 2.3.5. Suppose F is a family of sets. Then the intersection and
union of F are the sets ∩F and ∪F defined as follows:
∩F ={x |∀A ∈ F(x ∈ A)}={x |∀A(A ∈ F → x ∈ A)}.
∪F ={x |∃A ∈ F(x ∈ A)}={x |∃A(A ∈ F ∧ x ∈ A)}.