第二章
Fundamentals
of Logic
In
the development of any mathematical theory, assertions are made in
the form of sentences. Such verbal or written assertions, called
statements (or propositions), are declarative sentences that are
either true or false.
1)Transform
a given statement p into the statement ¬p,
which denotes its negation and is read “Not p.”
2)
a)
Conjunction:The conjunction of the statements p,q is denoted by p^q,
which is read “p and q.”
b)
Disjunction: The expression p V q denotes the disjunction of the
statements p,q and is read “p or q.”
c)Implication:
We say that “p implies q” and write p->q to designate the
statement, which is the implication of q by p.
i)If
p, then q.
ii)p
is sufficient for q
iii)p
is sufficient condition for q
iv)q
is necessary for p
v)q
is a necessary condition for p
vi)
p only if q
d)
Bi-conditional: The bi-conditional of two state¬ments p,q is
denoted by
p
↔ q, which is read “p if and only if q” or “p is necessary
and sufficient for q”
“p
if and only if q” 可縮寫成
“p
iff q”
p q
p^q p V q p V
q p->q q ↔ p
0 0 0 0 0 1 1
0 1 0 1 1 1 0
1 0 0 1 1 0 0
1 1 1 1 0 1¬ 1
A
compound statement is called a tautology if it is true for all truth
value assignments for its component statements. If a compound
statement is false for all such assignments, then it is called a
contradiction.
¬
Two
statements s1,s2 are said to be logically equivalent, and we write
s1
↔ s2, when the statement s1 is true if and only if the statement s2
is true.
¬
Let
s be a statement. If s contains no logical connectives other than ^
and V, then the dual of s denoted sd, is the statement obtained from
s by replacing each occurrence of ^ and V by V and ^, respectively,
and each occurrence of T0 and F0 by F0 and To, respectively.
The
principle of duality, let s and t be statements that contain no
logical connectives other than ^ and V.
if
s ↔ t, than sd ↔ td
p->q
↔ ~p V q
The
statement ¬q → ¬p is called contrapositive of the implication p
→ q
The
statement q → p is called the converse of p → q
¬p
→ ¬q is called the inverse of p → q
The
contrapositive(否逆)：¬q
→ ¬p
The
converse(逆): q
→ p
The
inverse (否)
: ¬p → ¬q
When
we write allxp(x)->exist x p(x) we are saying that the implication
allxp(x) → exist x p(x) is a logical implication.
Two substitution rules
1)Suppose that the compound statement P is a tautology. If p is primitive statement that appears in P and we replace each occurrence of p by the same q, then the resulting compound statement P1 is also a taotology.
2)Let P be a compound statement where p is an arbitrary statement that appears in P, and let q be a statement such that q ↔ p, Suppose that in P we replace one or more occurences of p by q. Then this replacement yields the compound statement P1. Under these circumstances P1 ↔ P.
p → q <=> ~p V q <=> ~q → ~p
Two substitution rules
1)Suppose that the compound statement P is a tautology. If p is primitive statement that appears in P and we replace each occurrence of p by the same q, then the resulting compound statement P1 is also a taotology.
2)Let P be a compound statement where p is an arbitrary statement that appears in P, and let q be a statement such that q ↔ p, Suppose that in P we replace one or more occurences of p by q. Then this replacement yields the compound statement P1. Under these circumstances P1 ↔ P.
p → q <=> ~p V q <=> ~q → ~p
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