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Logic
J.M.Basilla
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Math 1
General Mathematics
Lecture 7
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Introduction to Logic
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Institute of Mathematics
University of the Philippines-Diliman
[email protected]
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Julius Magalona Basilla
2012 Math 1
Statements
Propositional Logic
Negation
Conditional
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Some quotes
Logic
J.M.Basilla
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Statements
A doctor can bury his mistakes but an architect
can only advise his client to plant vine.
Frank Lloyd Right
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When a mathematican makes a mistake, he
simply erases the board.
Propositional Logic
Negation
Conditional
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The nature of mathematcs
Logic
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A mathematician wants to be sure that a certain
assertion actually follows from what has already
been accepted.
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Computations that accompany mathematics aid in
demonstrating these two points.
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Equally as important, he wants to be sure that a
certain assertion does not follow logically from the
other.
Statements
Propositional Logic
Negation
Conditional
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The nature of mathematcs
Logic
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Statements
In some sense, mathematics is just an applied
logic.
Propositional Logic
Negation
Conditional
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It is a pity that every human were given the ability to
think, but not everyone have the ability to reason
correctly.
Logic is the science of correct reasoning.
Logic helps us decide if an argument is valid or not.
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Example If there are fewer cars on the
roads the pollution will be acceptable.
Either we have fewer cars on the road or
there should be road pricing, or both. If
there is road pricing the summer will be
unbearably hot. The summer is actually
turning out to be quite cool. The conclusion
is inescapable: pollution is acceptable.
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Logic
Logic
J.M.Basilla
Statements
Propositional Logic
Negation
Conditional
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Logic
Logic
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When determining whether an argument is valid or
not, we have to focus on the form of the argument;
devoid of the meaning, particularly the emotional
loads that go with the sentences.
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Statements
Propositional Logic
Negation
Conditional
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Statements
Logic
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Statements
Non statements
Open Sesame.
Uranus has 65 rings.
This statement is false.
All men are created equal.
What country hosted the
recent olympics?
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1+2=3
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Today is a holiday.
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1. By a statement, we mean a declarative sentence
which can be categorically classified as true or false.
Good Luck!
3. Statements will be represented by single letters say
p,q,r, etc.
J.M.Basilla
Statements
Propositional Logic
Negation
Conditional
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Some english sentences which are not a
mathematical statements
A caveat!
Logic
J.M.Basilla
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Statements
Not all english sentences are mathematical
statement.
” I am lying now.” is not a mathematical statement.
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”This statement is false.” is also not a mathematical
statement.
Propositional Logic
Negation
Conditional
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Negation of a statement
Logic
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Hence, if p is true, its negation, denoted by ∼ p, is
false.
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The negation of a statement p is the statement
whose meaning is exactly the opposite of p.
Likewise, if p is false then ∼ p is true.
Juan is running.
Juan is not running.
One foot is equal to 10
inches.
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Jane did not pass Math I
last semester.
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Negation ∼ p
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Statement p
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Some easy example
Jane passed Math I last
semester.
One foot is not equal to 10
inches.
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Statements
Propositional Logic
Negation
Conditional
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Negation of statements involving quantifier
Logic
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All students are diligent
Some students are not
diligent
Nobody stole the cookie
from the cookie jar
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Some people are funny
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Negation ∼ p
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Statement p
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Example
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Forming the negation of a statement p which
contains quantifiers such as all, some, none or no is
not as easy as the previous examples.
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J.M.Basilla
No person is funny.
Somebody
stole
the
cookie from the cookie jar
Statements
Propositional Logic
Negation
Conditional
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General rule for negating statements
involving quantifier
Logic
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Statements
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Negation ∼ p
Some . . . not . . ..
None/No/No
one/Nobody. . ..
Some . . ..
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Some . . ..
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All/every . . ..
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Some . . . not . . ..
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All/Every . . ..
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Statement p
No/None . . ..
Propositional Logic
Negation
Conditional
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More example and exercises
Logic
J.M.Basilla
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My car did start.
Some of the cars did start.
Every car started.
Some of the cars did not
start.
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Some of the cars started.
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None of the car started.
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All the cars started.
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Some of the cars did not
start.
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My car did not start.
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Negation
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Statement
No car started.
Statements
Propositional Logic
Negation
Conditional
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Logic
∼p
F
T
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Statements
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p
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The truth of a negation
Propositional Logic
Negation
Conditional
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Conditional Statements
Logic
J.M.Basilla
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The statement P , is called the hypothesis or
antecedent of the conditional statement.
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A conditional statement is usually stated in the form ”
If P then Q.”
The statement Q is called the conclusion or
consequent of the conditional statement.
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A conditional statement ”If P then Q.” asserts that Q
becomes true the moment P becomes true.
However, it does not assert anything if P is false.
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In symbol, ”If P then Q.” is written as P → Q.
An example:
If you study hard, then you will graduate with honors.
Statements
Propositional Logic
Negation
Conditional
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Logic
J.M.Basilla
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Statements
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p =⇒ q
T
F
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T
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q
T
F
T
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p
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The truth of a conditional
Propositional Logic
Negation
Conditional
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The different form of conditional statements
Logic
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If you study hard then you will graduate
with honors.
P implies
Q.
Studying hard implies that you will graduate with honors.
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All P are Q.
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If P then Q.
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P : You study hard.
Q : You will graduate with honors.
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All those who study hard graduate with
honors.
Statements
Propositional Logic
Negation
Conditional
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Converse
Inverse
Contrapositive
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Statements related to A → B.
Logic
B→A
∼ A →∼ B
∼ B →∼ A
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Let
P
Q
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If it is an IBM PC then it is a computer. True
= It is an IBM PC.
= It is a computer.
The given conditional is of the form P → Q.
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If it is a computer then it is an IBM PC.
False
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Converse
Inverse
If it is not an IBM PC then it is not a
computer. False
Contrapositive
If it is not a computer then it is not an
IBM PC. True
Statements
Propositional Logic
Negation
Conditional
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Logic
B→A
∼ A →∼ B
∼ B →∼ A
J.M.Basilla
Let
P
Q
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If x is an even number then the last digit of x is
2. False
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Converse
Inverse
Contrapositive
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Statements related to A → B.
= x is even.
= The last digit of x is 2.
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If the last digit of x is 2 then x is even.
True
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Converse
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The given conditional is of the form P → Q.
Inverse
If x is not even then the last digit of x
is not 2. True
Contrapositive
If the last number of x is not 2 then x
is not even. False
Statements
Propositional Logic
Negation
Conditional
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Converse
Inverse
Contrapositive
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Statements related to A → B.
Logic
B→A
∼ A →∼ B
∼ B →∼ A
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Let
P
Q
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All students are diligent individuals. False
= x is a student.
= x is diligent.
The given conditional is of the form P → Q.
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All diligent individuals are students.
False
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Converse
Inverse
All none students are not diligent.
False
Contrapositive
All non-diligent individuals are nonstudents. False
Statements
Propositional Logic
Negation
Conditional
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Logic
B→A
∼ A →∼ B
∼ B →∼ A
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Let
P
Q
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If two lines are perpendicular then two lines form
a right angle. True
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Converse
Inverse
Contrapositive
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Statements related to A → B.
= Two lines are perpendicular.
= Two lines form a right angle.
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If two lines form a right angle then the
two lines are perpendicular True
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Converse
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The given conditional is of the form P → Q.
Inverse
If two lines are not perpendicular then
they do not form a right angle. true
Contrapositive
If two lines do not form a right angle
Statements
Propositional Logic
Negation
Conditional
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Equivalent conditional statements
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The truth value of the inverse and the converse are
always the same.
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The converse is the contrapositive of the inverse.
conditional If it is an IBM PC then it is a true
computer.
contrapositive If it is not a computer then it is true
not an IBM PC.
inverse
If it is not an IBM PC then it is false
not a computer.
converse If it is a computer then it is an false
IBM PC.
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The truth value of the conditional and the
contrapositive are always the same
Logic
J.M.Basilla
Statements
Propositional Logic
Negation
Conditional
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Equivalent conditional statements
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The truth value of the inverse and the converse are
always the same.
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The converse is the contrapositive of the inverse.
conditional If x is an even number then the false
last digit of x is two.
false
contrapositive If the last digit of x is two then x
is even.
inverse
If x is not an even number then true
the last digit of x is not two.
converse If the last digit of x is two then x true
is even
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The truth value of the conditional and the
contrapositive are always the same
Logic
J.M.Basilla
Statements
Propositional Logic
Negation
Conditional
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false
false
Logic
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Statements
false
Propositional Logic
Negation
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conditional All students are diligent.
contrapositive All those not diligent are not students.
inverse
All none students are not diligent.
converse All those diligent are students.
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Biconditional statements
Conditional
false
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Logic
true
J.M.Basilla
true
Statements
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Propositional Logic
Negation
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conditional If two lines are perpendicular
they form a right angle.
contrapositive If two lines do not form a right
angle they are not perpendicular.
inverse
If two lines are not perpendicular then they do not form a right
angle.
converse If two lines form a right angle
then they are perpendicular.
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Biconditional statements
Conditional
true
true
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Biconditional statements
In symbol, P ↔ Q.
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These type of conditional are called biconditional.
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In the last two examples, the conditional, inverse,
converse and contrapositive all have the same truth
value.
Read as, P if and only if Q.
Equivalently, P and Q are equivalent.
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Used in definitions.
Logic
J.M.Basilla
Statements
Propositional Logic
Negation
Conditional
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Statements
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p ⇐⇒ q
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q
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p
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The truth of a biconditional
Propositional Logic
Negation
Conditional