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

2 The Nuts and Bolts of Proof, Third Edition
To learn how to read and understand proofs (this term will be defined
more precisely in the next few paragraphs) already written in a textbook and
to learn how to construct proofs on our own, we will proceed by breaking
them down into a series of simple steps and looking at the clues that lead
from one step to the next. "Logic" is the key that will help us in this process.
We will use the words "logic" and "logical" according to the definition
suggested by Irving Copi: "Logic is the study of methods and principles used
to distinguish good (correct) from bad (incorrect) reasoning."
Before we start, though, we need to know the precise meaning of some of
the most common words that appear in mathematics and logic books.
Statement: A statement is a sentence expressed in words (or mathematical
symbols) that is either true or false. Statements do not include exclamations,
questions, or orders. A statement cannot be true and false at the same time,
although it can be true or false when considered in different contexts. For
example, the statement "No man has ever been on the Moon" was true in
1950, but it is now false. A statement is simple when it cannot be broken
down into other statements {e.g., "It will rain." "Two plus two equals four."
"I Hke that book."). A statement is composite when it contains several simple
statements connected by punctuation and/or words such as and, although, or,
thus, then, therefore, because, for, moreover, however, and so on {e.g., "It will
rain, although now it is only windy." "I like that book, but the other one is
more interesting." "If we work on this problem, we will understand it
better.").
Hypothesis: A hypothesis is a statement that it is assumed to be true, and
from which some consequence follows. (For example, in the sentence "If we
work on this problem, we will understand it better" the statement "we work
on this problem" is the hypothesis.) There are other common uses of the
word hypothesis in other scientific fields that are considerably different from
the one listed here. For example, in mathematics, hypotheses are never
tested. In other fields {e.g., statistics, biology, psychology), scientists discuss
the need "to test the hypothesis."
Conclusion: A conclusion is a statement that follows as a consequence from
previously assumed conditions (hypotheses). (For example, in the sentence
"If we work on this problem, we will understand it better" the statement "we
will understand it better" is the conclusion.) In The Words of Mathematics,
Steven Schwartzman writes, "In mathematics, the conclusion is the 'closing'
of a logical argument, the point at which all the evidence is brought together
and a final result obtained."
Definition: A definition is an unequivocal statement of the precise meaning of
a word or phrase, a mathematical symbol or concept, ending all possible
confusion. Definitions are Hke the soil in which a theory grows, and it is