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

Introduction and Basic Terminology 25
numbers listed. Moreover, q is not divisible by any of the prime numbers
because the quotient:
q 1
— = P1P2 • • • Pk-lPk+l "'Pn-\
Pk
Pk
is not an integer. This implies that ^ is a prime number because it is not
divisible by any prime number. But, we had assumed that we had a complete
hst of prime numbers; therefore, the collection P of all prime numbers is
infinite. •
HOW TO CONSTRUCT THE NEGATION OF A
STATEMENT
The truth of some statements can be proved in more than one way,
either by using direct proof or using proof by contrapositive. Generally, if
we have a choice, we should use direct proof. Indeed, direct proofs are
usually more intuitively understood and more informative. Moreover,
when using proof by contrapositive, there is an important point that needs
to be addressed. We have to construct the statement "not B" to use as the
hypothesis, and this can be a tricky step. Sometimes it is enough to insert the
word not in B to achieve our goal, as it happens in the previous examples.
The statements "x + j is irrational" and "the collection is infinite" are
changed into the statements "x + y is not irrational" and "the collection is
not infinite."
Other cases are not so easy to handle, especially when B includes
words such as unique, for one, for all, every, and none. These expressions are
usually called quantifiers. Let us see how we can work with some of these
expressions.
Original Statement
At least one
Some
All objects in a collection
have a certain property
Every object in a collection
has a certain property
None
There is no
Negative
None
None
There is at least one object in the collection that
does not have that property
There is at least one object in the collection that
does not have that property
There is at least one
There is at least one