Page 42 - HOW TO PROVE IT: A Structured Approach, Second Edition

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28 Sentential Logic
and subjectivity into our notation that is best avoided in mathematical writing.
It is therefore usually better to define such a set by spelling out the pattern that
determines the elements of the set.
In this case we could be explicit by defining B as follows:

B ={x | x is a prime number}.

This is read “B = the set of all x such that x is a prime number,” and it means
that the elements of B are the values of x that make the statement “x is a prime
number” come out true. You should think of the statement “x is a prime number”
as an elementhood test for the set. Any value of x that makes this statement
come out true passes the test and is an element of the set. Anything else fails
the test and is not an element. Of course, in this case the values of x that make
the statement true are precisely the prime numbers, so this definition says that
B is the set whose elements are the prime numbers, exactly as before.

Example 1.3.2. Rewrite these set definitions using elementhood tests:
1. E ={2, 4, 6, 8,...}.
2. P = {George Washington, John Adams, Thomas Jefferson, James
Madison, . . . }.
Solutions
Although there might be other ways of continuing these lists of elements,
probably the most natural ones are given by the following definitions:
1. E ={n | n is a positive even integer}.
2. P ={z | z was a president of the United States}.
If a set has been defined using an elementhood test, then that test can be used
to determine whether or not something is an element of the set. For example,
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consider the set {x | x < 9}. If we want to know if 5 is an element of this set,
we simply apply the elementhood test in the definition of the set – in other
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words, we check whether or not 5 < 9. Since 5 = 25 > 9, it fails the test,
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2
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so 5 ∈{x | x < 9}. On the other hand, (−2) = 4 < 9, so −2 ∈{x | x < 9}.
The same reasoning would apply to any other number. For any number y, to
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determine whether or not y ∈{x | x < 9}, we just check whether or not y < 9.
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In fact, we could think of the statement y ∈{x | x < 9} as just a roundabout
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way of saying y < 9.
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Notice that because the statement y ∈{x | x < 9} means the same thing as
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y < 9, it is a statement about y, but not x! To determine whether or not y ∈
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{x | x < 9} you need to know what y is (so you can compare its square to 9), but
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not what x is. We say that in the statement y ∈{x | x < 9}, y is a free variable,