166 lines
4 KiB
Markdown
166 lines
4 KiB
Markdown
Page 61
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**Definition**
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A **statement** (or **proposition**) is a sentence that is true or false but not
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both.
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---
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Page 63
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**Definition**
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If $p$ is a statement variable, the **negation** of $p$ is "not $p$" or "It is
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not the case that $p$" and is denoted $\neg p$. It has opposite truth value from
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$p$: if $p$ is true, $\neg p$ is false; if $p$ is false, $\neg p$ is true.
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---
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Page 64
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**Definition**
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If $p$ and $q$ are statement variables, the **conjunction** of $p$ and $q$ is
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"$p$ and $q$", denoted $p \wedge q$. It is true when, and only when, both $p$
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and $q$ are true. If either $p$ or $q$ is false, or if both are false,
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$p \wedge q$ is false.
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---
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Page 64
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**Definition**
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If $p$ and $q$ are statement variables, the **disjunction** of $p$ and $q$ is
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"$p$ or $q$", denoted $p \vee q$. It is true when either $p$ is true, or $q$ is
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true, or both $p$ and $q$ are true; it is false only when both $p$ and $q$ are
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false.
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---
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Page 65
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**Definition**
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A **statement form** (or **propositional form**) is an expression made up of
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statement variables (such as $p$, $q$, and $r$) and logical connectives (such as
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$\neg$, $\wedge$, and $\vee$) that becomes a statement when actual statements
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are substituted for the component statement variables. The **truth table** for a
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given statement form displays the truth values that correspond to all possible
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combinations of truth values for its component statement variables.
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---
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Page 67
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**Definition**
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Two _statement forms_ are called **logically equivalent** if, and only if, they
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have identical truth values for each possible substitution of statements for
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their statement variables. The logical equivalence of statements forms $P$ and
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$Q$ is denoted by writing $P \equiv Q$.
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Two _statements_ are called **logically equivalent** if, and only if, they have
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logically equivalent forms when identical component statement variables are used
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to replace identical component statements.
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---
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Page 69
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**De Morgan's Laws**
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The negation of an _and_ statement is logically equivalent to the _or_ statement
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in which each component is negated.
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The negation of an _or_ statement is logically equivalent to the _and_ statement
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in which each component is negated.
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---
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Page 71
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**Definition**
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A **tautology** is a statement form that is always true regardless of the truth
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values of the individual statements substituted for its statement variables. A
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statement whose form is a tautology is a **tautological statement**.
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A **contradiction** is a statement form that is always false regardless of the
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truth values of the individual statements substituted for its statement
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variables. A statement whose form is a contradiction is a **contradictory
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statement**.
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---
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Page 72
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**Theorem 2.1.1 Logical Equivalences**
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Given any statement variables $p$, $q$, and $r$, a tautology $\mathbf{t}$ and a
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contradiction $\mathbf{c}$, the following logical equivalences hold.
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1. _Communitative laws:_
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$$ p \wedge q \equiv q \wedge p $$
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$$ p \vee q \equiv q \vee p $$
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2. _Associative laws:_
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$$ (p \wedge q) \wedge r \equiv p \wedge (q \wedge r) $$
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$$ (p \vee q) \vee r \equiv p \vee (q \vee r) $$
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3. _Distributive laws:_
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$$ p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r) $$
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$$ p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r) $$
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4. _Identity laws:_
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$$ p \wedge \mathbf{t} \equiv p $$
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$$ p \vee \mathbf{c} \equiv p $$
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5. _Negation laws:_
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$$ p \vee \neg p \equiv \mathbf{t} $$
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$$ p \wedge \neg p \equiv \mathbf{c} $$
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6. _Double negative law:_
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$$ \neg(\neg p) \equiv p $$
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6. _Idempotent laws:_
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$$ p \wedge p \equiv p $$
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$$ p \vee p \equiv p $$
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8. _Universal bound laws:_Double
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$$ p \vee \mathbf{t} \equiv \mathbf{t} $$
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$$ p \wedge \mathbf{c} \equiv \mathbf{c} $$
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9. _De Morgan's laws:_
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$$ \neg (p \wedge q) \equiv \neg p \vee \neg q $$
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$$ \neg (p \vee q) \equiv \neg p \wedge \neg q $$
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10 _Absorption laws:_
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$$ p \vee (p \wedge q) \equiv p $$
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$$ p \wedge (p \vee q) \equiv p $$
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11. _Negations of $\mathbf{t}$ and $\mathbf{c}$:_
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$$ \neg \mathbf{t} \equiv \mathbf{c} $$
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$$ \neg \mathbf{c} \equiv \mathbf{t} $$
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