🚧 Setup for 8.2

chapter_8/notes.md

@@ -26,3 +26,73 @@ $A_1 \times A_2 \times \cdots \times A_n$ is a subset of
 $A_1 \times A_2 \times \cdots \times A_n$. The special cases of $2$-ary,
 $3$-ary, and $4$-ary relations are called **binary**, **ternary**, and
 **quarternary relations**, respectively.
+
+---
+
+Page 518
+
+**Definition**
+
+Let $R$ be a relation on a set $A$.
+
+1. $R$ is **reflexive** if, and only if, for every $x \in A, x R x$.
+
+2. $R$ is **symmetric** if, and only if, for every
+   $x, y \in A, \text{ if } x R y \text{ then } y R x$.
+
+3. $R$ is **transitive** if, and only if, for every
+   $x, y, z \in A, \text{ if } x R y \text{ and } y R z \text{ then } x R z$.
+
+---
+
+Page 523
+
+**Proof of Reflexivity:**
+
+Suppose $m$ is a particular but arbitrarily chosen integer. _[We must show that
+$m T m$.]_ Now $m - m = 0$. But $3 | 0$ since $0 = 3 \cdot 0$. Hence
+$3 | (m - m)$. Thus, by definition of $T$, $m T m$ _[as was to be shown]_.
+
+---
+
+Page 524
+
+**Proof of Symmetry:**
+
+Suppose $m$ and $n$ are particular but arbitrarily chosen integers that satisfy
+the condition $m T n$. _[We must show that $n T m$.]_ By definition of $T$,
+since $m T n$ then $3 | (m - n)$. By definition of "divides", this means that
+$m - n = 3k$, for some integer $k$. Multiplying both sides by $-1$ gives
+$n - m = 3(-k)$. Since $-k$ is an integer, this equation shows that
+$3 | (n - m)$. Hence, by definition of $T$, $n T m$ _[as was to be shown]_.
+
+---
+
+Page 524
+
+**Proof of Transitivity:**
+
+Suppose $m$, $n$, and $p$ are particular but arbitrarily chosen integers that
+satisfy the condition $m T n$ and $n T p$. _[We must show that $m T p$.]_ By
+definition of $T$, since $m T n$ and $n T p$, then $3 | (m - n)$ and
+$3 | (n - p)$. By definition of "divides", this means that $m - n = 3r$ and
+$n - p = 3s$, for some integers $r$ and $s$. Adding the two equations gives
+$(m - n) + (n - p) = 3r + 3s$, and simplifying gives that $m - p = 3(r + s)$.
+Since $r + s$ is an integer, this equation shows that $3 | (m - p)$. Hence, by
+definition of $T$, $m T p$ _[as was to be shown]_.
+
+---
+
+Page 525
+
+**Definition**
+
+Let $A$ be a set and $R$ a relation on $A$. The **transitive closure** of $R$ is
+the relation $R^t$ on $A$ that satisfies the following three properties:
+
+1. $R^t$ is transitive.
+
+2. $R \subseteq R^t$.
+
+3. If $S$ is any other transitive relation that contains $R$, then
+   $R^t \subseteq S$.