104 lines
2.8 KiB
Markdown
104 lines
2.8 KiB
Markdown
**Test Yourself**
|
|
|
|
Page 194
|
|
|
|
1. An integer is even if, and only if, ______.
|
|
|
|
it equals twice some integer.
|
|
|
|
2. An integer is odd if, and only if, ______.
|
|
|
|
it equals twice some integer plus 1.
|
|
|
|
3. An integer $n$ is prime if, and only if, ______.
|
|
|
|
$n$ is greater than $1$ and if $n$ equals the product of any two positive
|
|
integers, then one of the integers equals $1$ and the other equals $n$.
|
|
|
|
4. The most common way to disprove a universal statement is to find ______.
|
|
|
|
a counterexample.
|
|
|
|
5. According to the method of generalizing from the generic particular, to show
|
|
that every element of a set satisfies a certain property, suppose $x$ is a
|
|
______, and show that ______.
|
|
|
|
particular but arbitrarily chosen element of the set; $x$ satisfies the given
|
|
property.
|
|
|
|
6. To use the method of direct proof to prove a statement of the form, "For
|
|
every $x$ in a set $D$, if $P(x)$ then $Q(x)$," one supposes that ______ and
|
|
one shows that ______.
|
|
|
|
$x$ is a particular but arbitrarily chosen element of the set $D$ that makes the
|
|
hypothesis $P(x)$ true; $x$ makes the conclusion $Q(x)$ true.
|
|
|
|
---
|
|
|
|
**Test Yourself**
|
|
|
|
Page 204
|
|
|
|
1. The meaning of every variable used in a proof should be explained with
|
|
______.
|
|
|
|
The body of the proof.
|
|
|
|
2. Proofs should be written in sentences that are ______ and ______.
|
|
|
|
complete; grammatically correct
|
|
|
|
3. Every assertion in a proof should be supported by a ______.
|
|
|
|
reason
|
|
|
|
4. The following are some useful "little words and phrases" that clarify the
|
|
reasoning in a proof:
|
|
|
|
______, ______, ______, ______, and ______.
|
|
|
|
because; since; then; thus; so; hence; therefore; consequently; it follows that;
|
|
by substitution
|
|
|
|
5. A new thought or fact that does not follow as an immediate consequence of the
|
|
preceding statement can be introduced by writing ______, ______, ______,
|
|
______, or ______.
|
|
|
|
observe that; note that; recall that; but; now
|
|
|
|
6. To introduce a new variable that is defined in terms of previous variables,
|
|
use the word ______.
|
|
|
|
let
|
|
|
|
7. Displaying equations and inequalities increases the ______ of a proof.
|
|
|
|
readability
|
|
|
|
8. Some proof-writing mistakes are ______, ______, ______, ______, ______,
|
|
______, and ______.
|
|
|
|
arguing from examples; using the same letter to mean two different things;
|
|
jumping to a conclusion; assuming what is to be proved; confusion between what
|
|
is known and what is still to be shown; use of _any_ when the correct word is
|
|
_some_; misuse of the word _if_
|
|
|
|
---
|
|
|
|
**Test Yourself**
|
|
|
|
Page 210
|
|
|
|
1. To show that a real number is rational, we must show that we can write it as
|
|
______.
|
|
|
|
The ratio of integers, where the denominator is not 0.
|
|
|
|
2. An irrational number is a ______ that is ______.
|
|
|
|
real number; not rational
|
|
|
|
3. Zero is a rational number because ______.
|
|
|
|
zero is an integer that is a ratio of integers where the denominator is not
|
|
zero, $0 = \dfrac{0}{1}$.
|