7.3 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."
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Definition
An **argument is a sequence of statements, and an argument form is a
sequence of statement forms. All statements in an argument and all statement
forms in an argument form, except for the final one, are called premises (or
assumptions or hypotheses). The final statement or statement form is
called the conclusion. The symbol \therefore, which is read "therefore,"
is normally placed just before the conclusion.
To say that an argument form is valid means that no matter what particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true. To say that an argument is valid means that its form is valid.
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testing an Argument for Validity
-
Identify the premises and conclusion of the argument form.
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Construct a truth table showing the truth values of all the premises and the conclusion.
-
A row of the truth table in which all the premises are true is called a critical row. If there is a critical row in which the conclusion is false, then it is possible for an argument of the given form to have true premises and a false conclusion, and so the argument form is invalid. If the conclusion in every critical row is true, then the argument form is valid.
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Definition
An argument is called sound if, and only if, it is valid and all its premises are true. An argument that is not sound is called unsound.
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Contradiction Rule
If you can show that the supposition that statement p is false leads logically
to a contradiction, then you can conclude that $pr is true.