Page 88 - HOW TO PROVE IT: A Structured Approach, Second Edition

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74 Quantificational Logic
Similar notation is often used if the elements of a set have been numbered.
For example, suppose we wanted to form the set whose elements are the first
100 prime numbers. We might start by numbering the prime numbers, calling
them p 1 , p 2 , p 3 ,.... In other words,p 1 = 2, p 2 = 3, p 3 = 5, and so on. Then
the set we are looking for would be the set P ={p 1 , p 2 , p 3 ,..., p 100 }. Another
way of describing this set would be to say that it consists of all numbers p i , for
i an element of the set I ={1, 2, 3,..., 100}={i ∈ N | 1 ≤ i ≤ 100}. This
could be written P ={p i | i ∈ I}. Each element p i in this set is identified by
a number i ∈ I, called the index of the element. A set defined in this way is
sometimes called an indexed family, and I is called the index set.
Although the indices for an indexed family are often numbers, they need not
be. For example, suppose S is the set of all students at your school. If we wanted
to form the set of all mothers of students, we might let m s stand for the mother
of s, for any student s. Then the set of all mothers of students could be written
M ={m s | s ∈ S}. This is an indexed family in which the index set is S, the set
of all students. Each mother in the set is identified by naming the student who
is her child. Note that we could also define this set using an elementhood
test, by writing M ={m | m is the mother of some student} ={m |∃s ∈
S(m = m s )}. In general, any indexed family A ={x i | i ∈ I} can also be de-
fined as A ={x |∃i ∈ I(x = x i )}. It follows that the statement x ∈{x i | i ∈ I}
means the same thing as ∃i ∈ I(x = x i ).

Example 2.3.1. Analyze the logical forms of the following statements by
writing out the definitions of the set theory notation used.
√
1. y ∈{ x|x ∈ Q}.
3
2. {x i | i ∈ I}⊆ A.
3
2
3. {n | n ∈ N} and {n | n ∈ N} are not disjoint.
Solutions
√
1. ∃x ∈ Q(y = 3 x).
2. By the definition of subset we must say that every element of {x i | i ∈ I}
is also an element of A, so we could start by writing ∀x(x ∈{x i | i ∈ I}→
x ∈ A). Filling in the meaning of x ∈{x i | i ∈ I}, which we worked out
earlier, we would end up with ∀x(∃i ∈ I(x = x i ) → x ∈ A). But since the
elements of {x i | i ∈ I} are just the x i ’s, for all i ∈ I, perhaps an easier
way of saying that every element of {x i | i ∈ I} is an element of A would be
∀i ∈ I(x i ∈ A). The two answers we have given are equivalent, but showing
this would require the methods we will be studying in Chapter 3.