🚧 Setup for 4.1

chapter_4/notes.md

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