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**. --- Page 72 **Theorem 2.1.1 Logical Equivalences** Given any statement variables $p$, $q$, and $r$, a tautology $\mathbf{t}$ and a contradiction $\mathbf{c}$, the following logical equivalences hold. 1. _Communitative laws:_ $$ p \wedge q \equiv q \wedge p $$ $$ p \vee q \equiv q \vee p $$ 2. _Associative laws:_ $$ (p \wedge q) \wedge r \equiv p \wedge (q \wedge r) $$ $$ (p \vee q) \vee r \equiv p \vee (q \vee r) $$ 3. _Distributive laws:_ $$ p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r) $$ $$ p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r) $$ 4. _Identity laws:_ $$ p \wedge \mathbf{t} \equiv p $$ $$ p \vee \mathbf{c} \equiv p $$ 5. _Negation laws:_ $$ p \vee \neg p \equiv \mathbf{t} $$ $$ p \wedge \neg p \equiv \mathbf{c} $$ 6. _Double negative law:_ $$ \neg(\neg p) \equiv p $$ 6. _Idempotent laws:_ $$ p \wedge p \equiv p $$ $$ p \vee p \equiv p $$ 8. _Universal bound laws:_Double $$ p \vee \mathbf{t} \equiv \mathbf{t} $$ $$ p \wedge \mathbf{c} \equiv \mathbf{c} $$ 9. _De Morgan's laws:_ $$ \neg (p \wedge q) \equiv \neg p \vee \neg q $$ $$ \neg (p \vee q) \equiv \neg p \wedge \neg q $$ 10 _Absorption laws:_ $$ p \vee (p \wedge q) \equiv p $$ $$ p \wedge (p \vee q) \equiv p $$ 11. _Negations of $\mathbf{t}$ and $\mathbf{c}$:_ $$ \neg \mathbf{t} \equiv \mathbf{c} $$ $$ \neg \mathbf{c} \equiv \mathbf{t} $$ --- Page 77 **Definition** If $p$ and $q$ are statement variables, the **conditional** of $q$ by $p$ is "If $p$ then $q$" or "$p$ implies $q$" and is denoted $p \to q$. It is false when $p$ is true and $q$ is false; otherwise it is true. We call $p$ the **hypothesis** (or **antecedent**) if the conditional and $q$ the **conclusion** (or **consequent**). --- Page 80 **Definition** The **contrapositive** of a conditional statement of the form "If $p$ then $q" is $$ \text{If } \neg q \text{ then } \neg p $$ Symbolically, The contrapositive of $p \to q$ is $\neg q \to \neg p$. --- Page 81 **Definition** Suppose a conditional statement of the form "If $p$ then $q$" is given. 1. The **converse** is "If $q$ then $p$." 2. The **inverse** is "If $\neg p$ then $\neg q$." Symbolically, The converse of $p \to q$ is $q \to p$, and The inverse of $p \to q$ is $\neg p \to \neg q$. --- Page 82 **Definition** If $p$ and $q$ are statements, $p$ **only if** $q$ means "if not $q$, then not $p$," or, equivalently, "if $p$ then $q$." --- Page 83 **Definition** Given statement variables $p$ and $q$, the **biconditional of $p$ and $q$** is "$p$ if, and only if, $q$", and is denoted $p \leftrightarrow q$. It is true if both $p$ and $q$ have the same truth values and is false if $p$ and $q$ have opposite truth values. The words _if and only if_ are sometimes abbreviated **iff**. --- Page 84 **Definition** If $r$ and $s$ are statements: $r$ is a **sufficient condition** for $s$ means "if $r$ then $s$." $r$ is a **necessarily condition** for $s$ means "if not $r$, then not $s$."