🚧 Started chapter 2

chapter_2/notes.md

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+Page 61
+
+**Definition**
+
+A **statement** (or **proposition**) is a sentence that is true or false but not
+both.
+
+---
+
+Page 63
+
+**Definition**
+
+If $p$ is a statement variable, the **negation** of $p$ is "not $p$" or "It is
+not the case that $p$" and is denoted $\neg p$. It has opposite truth value from
+$p$: if $p$ is true, $\neg p$ is false; if $p$ is false, $\neg p$ is true.
+
+---
+
+Page 64
+
+**Definition**
+
+If $p$ and $q$ are statement variables, the **conjunction** of $p$ and $q$ is
+"$p$ and $q$", denoted $p \wedge q$. It is true when, and only when, both $p$
+and $q$ are true. If either $p$ or $q$ is false, or if both are false,
+$p \wedge q$ is false.
+
+---
+
+Page 64
+
+**Definition**
+
+If $p$ and $q$ are statement variables, the **disjunction** of $p$ and $q$ is
+"$p$ or $q$", denoted $p \vee q$. It is true when either $p$ is true, or $q$ is
+true, or both $p$ and $q$ are true; it is false only when both $p$ and $q$ are
+false.
+
+---
+
+Page 65
+
+**Definition**
+
+A **statement form** (or **propositional form**) is an expression made up of
+statement variables (such as $p$, $q$, and $r$) and logical connectives (such as
+$\neg$, $\wedge$, and $\vee$) that becomes a statement when actual statements
+are substituted for the component statement variables. The **truth table** for a
+given statement form displays the truth values that correspond to all possible
+combinations of truth values for its component statement variables.
+
+---
+
+Page 67
+
+**Definition**
+
+Two _statement forms_ are called **logically equivalent** if, and only if, they
+have identical truth values for each possible substitution of statements for
+their statement variables. The logical equivalence of statements forms $P$ and
+$Q$ is denoted by writing $P \equiv Q$.
+
+Two _statements_ are called **logically equivalent** if, and only if, they have
+logically equivalent forms when identical component statement variables are used
+to replace identical component statements.
+
+---
+
+Page 69
+
+**De Morgan's Laws**
+
+The negation of an _and_ statement is logically equivalent to the _or_ statement
+in which each component is negated.
+
+The negation of an _or_ statement is logically equivalent to the _and_ statement
+in which each component is negated.
+
+---
+
+Page 71
+
+**Definition**
+
+A **tautology** is a statement form that is always true regardless of the truth
+values of the individual statements substituted for its statement variables. A
+statement whose form is a tautology is a **tautological statement**.
+
+A **contradiction** is a statement form that is always false regardless of the
+truth values of the individual statements substituted for its statement
+variables. A statement whose form is a contradiction is a **contradictory
+statement**.