5.6 KiB
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Definition
A statement (or proposition) is a sentence that is true or false but not both.
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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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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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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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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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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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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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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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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.
- Communitative laws:
p \wedge q \equiv q \wedge p p \vee q \equiv q \vee p - Associative laws:
(p \wedge q) \wedge r \equiv p \wedge (q \wedge r) (p \vee q) \vee r \equiv p \vee (q \vee r) - 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) - Identity laws:
p \wedge \mathbf{t} \equiv p p \vee \mathbf{c} \equiv p - Negation laws:
p \vee \neg p \equiv \mathbf{t} p \wedge \neg p \equiv \mathbf{c} - Double negative law:
\neg(\neg p) \equiv p - Idempotent laws:
p \wedge p \equiv p p \vee p \equiv p - _Universal bound laws:_Double
p \vee \mathbf{t} \equiv \mathbf{t} p \wedge \mathbf{c} \equiv \mathbf{c} - 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 - 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).
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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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Definition
Suppose a conditional statement of the form "If p then $q$" is given.
-
The converse is "If
qthenp." -
The inverse is "If
\neg pthen\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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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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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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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."