🚧 Setup for 3.2

chapter_3/notes.md

@@ -58,3 +58,84 @@ Let $P(x)$ and $Q(x)$ be predicates and suppose the domain of $x$ is $D$.
 - The notation $P(x) \Leftrightarrow Q(x)$ means that $P(x)$ and $Q(x)$ have
   identical truth sets, or, equivalently,
   $\forall x, P(x) \leftrightarrow Q(x)$.
+
+---
+
+Page 145
+
+**Theorem 3.2.1 Negation of a Universal Statement**
+
+The negation of a statement of the form
+
+$$ \forall \text{ in } D, Q(x) $$
+
+is logically equivalent to a statement of the form
+
+$$ \exists \text{ in } D \text{ such that } \neg Q(x) $$
+
+Symbolically,
+
+$$ \neg(\forall x \in D, Q(x)) \equiv \exists x \in D \text{ such that } \neg Q(x) $$
+
+---
+
+Page 146
+
+**Theorem 3.2.2 Negation of an Existential Statement**
+
+The negation of a statement of the form
+
+$$ \exists \text{ in } D \text{ such that } Q(x) $$
+
+is logically equivalent to a statement of the form
+
+$$ \forall x \text{ in } D, \neg Q(x) $$
+
+Symbolically,
+
+$$ \neg(\exists x \in D \text{ such that } Q(x)) \equiv \forall x \in D, \neg Q(x) $$
+
+---
+
+Page 148
+
+**Negation of a Universal Conditional Statement**
+
+$$ \neg(\forall x, \text{ if } P(x) \text{ then } Q(x)) \equiv \exists x \text{ such that } P(x) \text{ and } \neg Q(x) $$
+
+$$ \neg(\forall x, P(x) \to Q(x)) \equiv \exists x, (P(x) \wedge \neg Q(x)) $$
+
+---
+
+Page 150
+
+**Definition**
+
+Consider a statement of the form
+$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$.
+
+1. Its **contrapositive** is the statement
+   $\forall x \in D, \text{ if } \neg Q(x) \text{ then } \neg P(x)$.
+
+2. Its **converse** is the statement
+   $\forall x \in D, \text{ if } Q(x) \text{ then } P(x)$.
+
+3. Its **inverse** is the statement
+   $\forall x \in D, \text{ if } \neg P(x) \text{ then } \neg Q(x)$.
+
+---
+
+Page 151
+
+**Definition**
+
+- "$\forall x, r(x)$ is a **sufficient condition** for $s(x)$" means
+  "$\forall x, \text{ if } r(x) \text{ then } s(x)$."
+
+- "$\forall x, r(x)$ is a **necessary condition** for $s(x)$" means
+  "$\forall x, \text{ if } \neg r(x) \text{ then } \neg s(x)$" or, equivalently,
+  "$\forall x, \text{ if } s(x) \text{ then } r(x)$."
+
+- "$\forall x, r(x)$ **only if** $s(x)$" means
+  "$\forall x, \text{ if } \neg s(x) \text{ then } \neg r(x)$" or, equivalently,
+  "$\forall x, \text{ if } r(x) \text{ then } s(x)$."