discrete_mathematics_with_applications/chapter_2/notes.md

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**Definition**

A **statement** (or **proposition**) is a sentence that is true or false but not
both.

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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.

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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.

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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.

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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.

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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.

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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.

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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**.

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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 $\mathbf{c}$:_

$$ \neg \mathbf{t} \equiv \mathbf{c} $$

$$ \neg \mathbf{c} \equiv \mathbf{t} $$

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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**).

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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 $\neg q \to \neg p$.

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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 $\neg p \to \neg q$.

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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 $q$."

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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**.

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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 $s$."