Page 90 - HOW TO PROVE IT: A Structured Approach, Second Edition

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76 Quantificational Logic
Any time you are working with some subsets of a set X, it may be helpful to
remember that all of these subsets of X are elements of P (X), by the definition
of power set. For example, if we let C be the set of all courses offered at your
school, then each of the sets C s from our previous example is a subset of C.
Thus, for each student s, C s ∈ P (C). This means that every element of the
family F ={C s | s ∈ S} is an element of P (C), so F ⊆ P (C).

Example 2.3.3. Analyze the logical forms of the following statements.

1. x ∈ P (A).
2. P (A) ⊆ P (B).
3. B ∈{P (A) | A ∈ F}.
4. x ∈ P (A ∩ B).
5. x ∈ P (A) ∩ P (B).

Solutions
1. By the definition of power set, the elements of P (A) are the subsets of A.
Thus, to say that x ∈ P (A) means that x ⊆ A, which we already know can
be written as ∀y(y ∈ x → y ∈ A).
2. By the definition of subset, this means ∀x(x ∈ P (A) → x ∈ P (B)). Now,
writing out x ∈ P (A) and x ∈ P (B) as before, we get ∀x[∀y(y ∈ x →
y ∈ A) →∀y(y ∈ x → y ∈ B)].
3. As before, this means ∃A ∈ F(B = P (A)). Now, to say that B = P (A)
means that the elements of B are precisely the subsets of A, or in other words
∀x(x ∈ B ↔ x ⊆ A). Filling this in, and writing out the definition of subset,
we get our final answer, ∃A ∈ F ∀x(x ∈ B ↔∀y(y ∈ x → y ∈ A)).
4. Asinthefirststatement,westartbywritingthisas∀y(y ∈ x → y ∈ A ∩ B).
Now, filling in the definition of intersection, we get ∀y(y ∈ x → (y ∈ A ∧
y ∈ B)).
5. By the definition of intersection, this means (x ∈ P (A)) ∧ (x ∈ P (B)).
Now, writing out the definition of power set as before, we get ∀y(y ∈ x →
y ∈ A) ∧∀y(y ∈ x → y ∈ B).

Note that for statement 5 in this example we first wrote out the definition
of intersection and then used the definition of power set, whereas in statement
4 we started by writing out the definition of power set and then used the
definition of intersection. As you learn the definitions of more mathematical
terms and symbols, it will become more important to be able to choose which
definition to think about first when working out the meaning of a complex
mathematical statement. A good rule of thumb is to always start with the
“outermost” symbol. In statement 4 in Example 2.3.3, the intersection symbol