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78 Quantificational Logic
Some mathematicians consider ∩F to be undefined if F = ∅. For an ex-
planation of the reason for this, see exercise 14. We will use the notation ∩F
only when F = ∅.
Notice that if A and B are any two sets and F ={A, B}, then ∩F = A ∩ B
and ∪F = A ∪ B. Thus, the definitions of intersection and union of a family
of sets are actually generalizations of our old definitions of the intersection and
union of two sets.

Example 2.3.6. Analyze the logical forms of the following statements.

1. x ∈∩F.
2. ∩F ⊆∪G.
3. x ∈ P (∪F).
4. x ∈∪{P (A) | A ∈ F}.
Solutions

1. By the definition of the intersection of a family of sets, this means ∀A ∈
F(x ∈ A), or equivalently, ∀A(A ∈ F → x ∈ A).
2. As we saw in Example 2.2.1, to say that one set is not a subset of an-
other means that there is something that is an element of the first but not
the second. Thus, we start by writing ∃x(x ∈∩F ∧ x /∈∪G). We have al-
ready written out what x ∈∩F means in solution 1. By the definition of the
union of a family of sets, x ∈∪G means ∃A ∈ G(x ∈ A), so x /∈∪G means
¬∃A ∈ G(x ∈ A). By the quantifier negation laws, this is equivalent to
∀A ∈ G(x /∈ A). Putting this all together, our answer is ∃x[∀A ∈
F(x ∈ A) ∧∀A ∈ G(x /∈ A)].
3. Because the union symbol occurs within the power set notation, we start by
writing out the definition of power set. As in Example 2.3.3, we get x ⊆∪F,
or in other words ∀y(y ∈ x → y ∈∪F ). Now we use the definition of union
to write out y ∈∪F as ∃A ∈ F(y ∈ A). The final answer is ∀y(y ∈ x →
∃A ∈ F(y ∈ A)).
4. This time we start by writing out the definition of union. According to
this definition, the statement means that x is an element of at least one of
the sets P (A), for A ∈ F. In other words, ∃A ∈ F(x ∈ P (A)). Inserting
our analysis of the statement x ∈ P (A) from Example 2.3.3, we get ∃A ∈
F∀y(y ∈ x → y ∈ A).
Writing complex mathematical statements in logical symbols, as we did in
the last example, may sometimes help you understand what the statements
mean and whether they are true or false. For example, suppose that we once
again let C s be the set of all courses that have been taken by student s.