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

6 The Nuts and Bolts of Proof, Third Edition
But what else do we know about triangles; that is, what implicit
information do we have? We can use any previously proven result,
not only about triangles but also, for example, about geometric
properties of Hues and angles in general (impHcit information).
The conclusion we want to reach is that "the internal angles of the
triangle are equal." Therefore, it will be extremely important to
know the definition of "internal angles of a triangle" as well.
2. Consider the following statement:
The number a is a nonzero real number.
The statement gives the following information:
i. the number a is different from zero (explicit information); and
ii. the number a is a real number (explicit information).
As mentioned in the preceding, the second fact implicitly states that
we can use all properties of real numbers and their operation that the
book has already mentioned or requires readers to know (imphcit
information). Sometimes the hypotheses, as stated, might contain
nonessential details, which are given for the sake of clarity.
EXAMPLES
1. Consider the triangle ABC.
2. Let A be the collection of all even numbers.
3. Let a be a nonzero real number.
The fact that the triangle is denoted as ABC is not significant. We can use
any three letters (or other symbols) to name the three vertices of the triangle.
In the same way, we can use any letter to denote the collection of all
even numbers and a nonzero real number. The most important thing is
consistency. If we used the letters A, B, and C to denote the vertices of
a triangle, then these letters will refer to the vertices any time they are
mentioned in that context, and they cannot be used to denote another
object.
Only after we are sure that we can identify the hypothesis and the
conclusion and that we understand the meaning of a theorem to be proved
can we go on to read, understand, or construct its proof (that is, a logical
argument that will establish how and why the theorem we are considering is
true). It is important to observe that a mathematical statement to be proved
does not exist in a vacuum, but it is part of a larger context; therefore, its
proof might change significantly, depending on the material previously