Page 89 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More Operations on Sets 75
3. We must say that the two sets have a common element, so one solution
3
2
is to start by writing ∃x(x ∈{n | n ∈ N}∧ x ∈{n | n ∈ N}). However, as
in the last statement, there is an easier way. An element common to the
two sets would have to be the square of some natural number and also the
cube of some (possibly different) natural number. Thus, we could say that
2
3
there is such a common element by saying ∃n ∈ N∃m ∈ N(n = m ). Note
2
3
that it would be wrong to write ∃n ∈ N(n = n ), because this wouldn’t
allow for the possibility of the two natural numbers being different. By the
3
2
way, this statement is true, since 64 = 8 = 4 , so 64 is an element of both
sets.
Anything at all can be an element of a set. Some interesting and useful ideas
arise when we consider the possibility of a set having other sets as elements.
For example, suppose A ={1, 2, 3}, B ={4}, and C = ∅. There is no reason
why we couldn’t form the set F ={A, B, C}, whose elements are the three
sets A, B, and C. Filling in the definitions of A, B, and C, we could write
this in another way: F ={{1, 2, 3}, {4}, ∅}. Note that 1 ∈ A and A ∈ F but
1 /∈ F. F has only three elements, and all three of them are sets, not numbers.
Sets such as F, whose elements are all sets, are sometimes called families of
sets.
It is often convenient to define families of sets as indexed families. For
example, suppose we again let S stand for the set of all students, and for each
student s we let C s be the set of courses that s has taken. Then the collection
of all of these sets C s would be an indexed family of sets F ={C s | s ∈ S}.
Remember that the elements of this family are not courses but sets of courses. If
we let t stand for some particular student Tina, and if Tina has taken Calculus,
English Composition, and American History, then C t ={Calculus, English
Composition, American History} and C t ∈ F, but Calculus /∈ F.
An important example of a family of sets is given by the power set of
a set.

Definition 2.3.2. Suppose A is a set. The power set of A, denoted P (A), is
the set whose elements are all the subsets of A. In other words,

P (A) ={x | x ⊆ A}.

For example, the set A ={7, 12} has four subsets: ∅, {7}, {12}, and {7, 12}.
Thus, P (A) ={∅, {7}, {12}, {7, 12}}. What about P (∅)? Although ∅ has
no elements, it does have one subset, namely ∅. Thus, P (∅) = {∅}. Note
that, as we saw in Section 1.3, {∅} is not the same as ∅.