Logic in mathematics is the study of deduction. Its purpose is to develop a system of methods and principles to evaluate the validity of arguments. Logic uses operators or connectives( /\ =AND),
and (\/ = OR) to connect
propositions together. When two propositions are connected by the operator /\
the resulting one is a compound proposition called conjunction. If they
are joined by the
operator \/ the resulting one
is also a compound proposition called disjunction. Compound propositions
can be called conditional propositions and are denoted: p à q (p and q are two propositions of the form if p then
q). There are compound propositions called biconditional
propositions and are denoted:
p«q (p and q are two
propositions of the form p if and only q).
In order to determine the truth values of compound
propositions, logic uses truth tables which
are arrangements of truth
values that show in every possible case how the truth value of a compound proposition
is determined by the truth values of its simple components.
|
P |
Q |
PÙQ |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
F |
|
F |
F |
F |
Truth table providing the truth value of a conjunction
proposition
P /\ Q
|
P |
Q |
PÚQ |
|
T |
T |
T |
|
T |
F |
T |
|
F |
T |
T |
|
F |
F |
F |
Truth table providing the truth value of a disjunction
proposition
P \/ Q
|
P |
Q |
P®Q |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
Truth table providing the truth value of a conditional
proposition
P®Q
|
P |
Q |
P«Q |
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
F |
|
F |
F |
T |
Truth table providing the truth value of a biconditional
proposition
P « q
The converse of a conditional proposition P ® Q may be written q ® p.
The
contrapositive of the conditional proposition P®Q may be written ~q ® ~p
Finally, logic is a tool to analyze proofs using
mathematical induction as a special proof technique
that allows to prove the truth of a sample plus a rule
about the unexamined cases that actually gives
information about every member of the class.
Applications
Logic operators AND, OR, NOT are used by search engines in special technique called Boolean
Searching. URL: http://www.nova.edu/library/biz/biz_bool.htm
Logic is used in database management, programming
language design, and artificial intelligence.
URL: http://www.rbjones.com/rbjpub/logic
1) Is there a formulation of mathematical induction called weak form of mathematical
induction?
In page 43(1.6.3), there is a reference
of a strong form of mathematical induction.
2) Are undefined terms explicitly defined or implicitly defined ?
I could not find an example of undefined terms which any kind of validity.