Transcript p^q - Google Sites

LOGICAL
REASONING
by
Melanie Espejo
What is LOGIC?
LOGIC is often defined as the
science of thinking and reasoning
correctly.
It was first studied extensively by Aristotle in
approximately 400 B.C. Some famous
contributors to the development of logic were
Leonhard Euler(1707-1783), George Boole(18151864), John Venn(1834-1923), and Bertrand
Russel(1872-1970).
Symbolic logic enables us to solve
complex problems. The symbols
in logic, like those in algebra, are
important aids to our thinking. In
preparing for any task, we must
first equip ourselves with the
proper tools required to do the
job. The first thing we must be
able to do is identify and
symbolize sentences. In logic we
concern ourselves only with those
sentences that are either true or
false, but not both.
STATEMENT is a
declarative sentence
that is either TRUE or
FALSE (but not both
true and false).
COMPOUND
STATEMENT is the
result of joining two
or more statements
with connective
words such as and,
or, not, and if/then.
When is a compound
statement true?
Let us study first the
various ways in which the
statement can be a
negation, conjunction,
disjunction, conditional or
any combination thereof.
Negation( ̴p)
The negation of a
statement is the denial
statement and is
represented by the
symbol ̴
It is frequently formed
by inserting the word
“not”.
Negation( ̴p)
EXAMPLE:
p: Today is Thursday.
̴p: Today is not
Thursday.
CONJUNCTION (p^q)
A conjunction consist of
two or more statements
connected by the word
and. We use the symbol
to represent the word
and; thus the
conjunction “p^q”
represents the compound
statement “p and q”.
CONJUNCTION (p^q)
EXAMPLE:
p: Logic is easy.
q: Algebra is difficult.
Therefore:
1. Logic is easy and Algebra
is difficult. (p^q)
2. Logic is easy and Algebra
is not difficult. (p^ ̴q)
DISJUNCTION(pvq)
A disjunction is consists
of two or more
statements connected
by the word or. We use
the symbol v to
represent the word or.
Thus, the disjunction
“pvq” represents the
compound statement
“p or q”.
DISJUNCTION(pvq)
EXAMPLE:
p: Gustav is honest.
q: Iraz is honest .
Therefore:
Gustav is honest or Iraz is
honest. (pvq)
CONDITIONALS (p→q)
Any statement of the
form “If p, then q” is
called a conditional (or
an implication); p is
called the hypothesis
(or premise) of the
conditional, and q is
called the conclusion
of the conditional.
CONDITIONALS
(p→q)
EXAMPLE:
P: I exercise regularly.
q: I am healthy.
Therefore:
If I execise regularly, then I
am healthy. (p→q)
Can simply be stated as:
“If I execise regularly, I am
healthy”. (p→q)
Alternatively, the
conditional “ If p, then q”
may be phrased as
“q if p”.
(“I am healthy if I exercise
regularly”)
The table below
summarizes the logical
connectives and
symbols discussed:
LOGICAL CONNECTIVES
STATEMENTS
SYMBOL
Negation
Conjunction
̴
^
Disjunction
v
Conditional
implication
→
READ AS...
not
and
(inclusive) or
If.... then...
OTHER INFOS THAT YOU NEED TO
REVIEW:
INDUCTIVE REASONING is a process
of observing data, recognizing
patterns, and making generalizations.
DEDUCTIVE REASONING is a kind of
reasoning from general to particulars. It
involves the stipulation of a specific
statement based on a general statement
that has been accepted to be true.
To deduce means to reason from
known facts. When you prove a
theorem, you are using deductive
reasoning- using existing structures to
deduce new parts of the structure. In
deductive reasoning, you assume that
the hypothesis is true, and then write
a series of statements that leads to
the conclusion. Each statement is
supported by a reason that justifies it.
The set of statements and reasons
consists the proof.
A simple syllogism is an argument
made up of three statements: a
major premise, a minor premise
(both of which are accepted as
true), and a conclusion.
Definition is a statement of the
meaning of a word, or term, or
phrase which made use of
previously defined terms.
Postulate is a statement which is
accepted as true without proof.
Theorem is any statement that can
be proved true.
Corollary is a statement that
follows easily from a previously
proved theorem.
The CONVERSE of the conditional
statement is formed by
interchanging the hypothesis and
conclusion. For instance, the
converse of p→q is q→p. The
converse may also be true or false.
EXAMPLES:
1. If m<A=45, then <A is acute.
Ans: The given statement is true
because 45<90. Its converse
“If <A is acute, then m<A=45” is
false. Some acute angles do
not measure 45.
EXAMPLES:
2. If m<B=90, then <B is a right
triangle.
Ans: Both are true.
EXAMPLES:
3. If today is Sunday, then it is a
weekend day.
Ans: The statement is true because
Sunday is a weekend day. The
converse “If it is a weekend day,
then it is a Sunday” is false
because Saturday (a
counterexample) is also a
weekend day.
INVERSE of the statement:
The inverse of an implication, “If p
then q” is “If not p, then not q”. In
symbols, the inverse of p→q is
̴p→ ̴q
EXAMPLE:
If two sides of a triangle are equal,
then the angles opposite those
sides are equal.
Ans: If two sides of a triangles are
unequal, then the angles
opposite those sides are
unequal.
CONTRAPOSITIVE (counterexample)
of the statement:
The contrapositive of an implication is
the converse of its inverse. To form
the contrapositive of a statement, we
interchange the hypothesis and the
conclusion and negate each. In
symbols, the contrapositive of p→q is
̴q→ ̴p
EXAMPLE:
If two angles are opposite angles of
a parallelogram, then they are
equal.
Ans: If two angles of a
parallelogram are unequal, then
they are not opposite angles.
Write the converse,
inverse, and
counterexample of the
statement below:
STATEMENT:
If two angles are
opposite angles of a
parallelogram, then they
are equal.
CONVERSE: If two angles of a
parallelogram are equal, then
they are opposite angles.
INVERSE: If two angles of a
parallelogram are not opposite
angles, then they are unequal.
CONTRAPOSITIVE: If two
angles are of a parallelogram
are unequal, then they are not
opposite angles.
TRUTH TABLES:
TABLE 1P
Q
P→Q
T
T
T
T
F
F
F
T
T
F
F
T
TABLE 2P
Q
P↔Q
T
T
T
T
F
F
T
F
F
T
F
F
SU
TABLE 3P
Q
̴Q
̴ QvP
( ̴ Q v P)→Q
T
T
F
T
T
F
F
T
T
F
TABLE 4P
Q
P^ Q
T
T
T
T
F
F
F
T
F
F
F
F
TABLE 5P
Q
Pv Q
T
T
T
T
F
T
F
T
T
F
F
F
TABLE 6P
Q
̴P
̴P^Q
T
F
F
F
TABLE 7P
Q
̴P
̴Q
̴P ^ ̴Q
̴( ̴P ^ ̴Q)
F
F
T
T
T
F
The Thinker (French: Le
Penseur) is a bronze and
marble sculpture by Auguste
Rodin held in the Musée
Rodin in Paris. It depicts a man
in sobermeditation battling
with a powerful internal
struggle. It is often used to
represent philosophy.
F
“ Creative minds have always been known
to survive any kind of bad training.”
-Anna Freud (1895-1982)
“ Exercise your mind as you
F
exercise your body”.
MAG-ARAL!