105 lines
3.1 KiB
Markdown
105 lines
3.1 KiB
Markdown
Page 184
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**Assumptions**
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- In this text we assume familiarity with the laws of basic algebra, which are
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listed in Appendix A.
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- We also use the three properties of equality: For all objects $A$, $B$, and
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$C$, (1) $A = A$, (2) if $A = B$, then $B = 1$, and (3) if $A = B$ and
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$B = C$, then $A = C$.
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- And we use the principle of substitution: For all objects $A$ and $B$, if
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$A = B$, then we may substitute $B$ whenever we have $A$.
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- In addition, we assume that there is no integer between $0$ and $1$ and that
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the set of all integers is closed under addition, subtraction, and
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multiplication. This means that sums, differences, and products of integers
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are integers.
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---
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Page 185
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**Definitions**
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An integer $n$ is **even** if, and only if, $n$ equals twice some integer. An
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integer $n$ is **odd** if, and only if, $n$ equals twice some integer plus $1$.
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Symbolically, for any integer $n$
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$$ n \text{ is even} \Leftrightarrow n = 2k \text{ for some integer } k $$
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$$ n \text{ is odd} \Leftrightarrow n = 2k + 1 \text{ for some integer } k $$
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---
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Page 186
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**Definition**
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An integer $n$ is **prime** if, and only if, $n > 1$ and for all positive
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integers $r$ and $s$, if $n = rs$, then either $r$ or $s$ equals $n$. An integer
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$n$ is **composite** if, and only if, $n > 1$ and $n = rs$ for some integers $r$
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and $s$ with $1 < r < n$ and $1 < s < n$.
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In symbols: For each integer $n$ with $n > 1$,
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$$ n \text{ is prime} \Leftrightarrow \forall \text{ positive integers } r \text{ and } s, \text{ if } n = rs \text{ then either } r = 1 \text{ and } s = n \text{ or } r = n \text{ and } s = 1 $$
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$$ n \text{ is composite} \Leftrightarrow \exists \text{ positive integers } r \text{ and } s \text{ such that } n = rs \text{ and } 1 < r < n \text{ and } 1 < s < n $$
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---
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Page 188
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**Disproof by Counterexample**
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To disprove a statement of the form
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"$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$," find a value of $x$ in
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$D$ for which the hypothesis $P(x)$ is true and the conclusion $Q(x)$ is false.
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Such an $x$ is called a **counterexample**.
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---
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Page 189
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**Generalizing from the Generic Particular**
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To show that _every_ element of a set satisfies a certain property, suppose $x$
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is a _particular_ but _arbitrarily chosen_ element of the set, and show that $x$
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satisfies the property.
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Page 191
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**Existential Instantiation**
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If the existence of a certain kind of object is assumed or has been deduce, then
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it can be given a name, as long as that name is not currently being used to
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refer to something else in the same discussion.
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---
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Page 192
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**Theorem 4.1.1**
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The sum of any two even integers is even.
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**Proof:** Suppose $m$ and $n$ are any _[particular but arbitrarily chosen]_
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even integers. _[We must show that $m + n$ is even.]_ By definition of even,
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$m = 2r$ and $n = 2s$ for some integers $r$ and $s$. Then
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$$ m + n = 2r + 2s \quad \text{ by substitution} $$
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$$ \quad = 2(r + s) \quad \text{ by factoring out a 2} $$
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Let $t = r + s$. Note that $t$ is an integer because it is a sum of integers.
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Hence
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$$ m + n = 2r \quad \text{where } t \text{ is an integer} $$
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It follows by definition of even that $m + n$ is even. _[This is what we needed
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to show.]_
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