epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 69
considering—the universal set {e.g., integer numbers, such as cars produced
during January 2005 in Detroit plants). The second part (if needed) follows
the expression "such that" (usually represented by a vertical segment "|") and
lists additional properties. So, the description of a set might look Uke:
A ={n G ZI the remainder of the division of n by 2 is zero}
B ={cars produced during January 2005 in Detroit
plants I they have four doors}
Usually, in a general setting the universal set is represented by U.
A set A is contained in a set B (or ^4 is a subset of B) if every element of A is
an element of B. In this case, we write A C.B.
By definition of subset, every nonempty set has two trivial subsets, itself
and 0.
Two sets, A and B, are equal if the following two conditions are true:
1. A^B
2. BOA
The first of the two conditions states that every element of A is an element
of B. The second states that every element of B is an element of A. Therefore,
A and B have exactly the same elements.
EXAMPLE 1. Let A = {n eZ\ the remainder of the division of n by 2 is
zero} and J5 = {all integer multiples of 2}. Prove that A = B.
Proof:
Parti. A^B
Let X be a generic element of A (that is, x is any number satisfying the
conditions to belong to the set A). We need to prove that x is an element ofB
as well.
As X is an element of A, we know that:
2 = ^ + 0
where q is an integer number.
Thus, x = 2q. This means that x is a multiple of 2; therefore, x is an
element of B.
Part 2. B QA
Let X be an element of B. We need to prove that x is an element of A as
well.