Math 1
J.M.Basilla
Statements
Conditional
Statements
Logic
Statements related to a
conditional
Biconditional Statements
General Education Mathematics
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Julius Magalona Basilla
Department of Mathematics
University of the Philippines-Diliman
[email protected]
2008 Math 1 Lesson 4
Some quotes
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
A doctor can bury his mistakes but an architect
can only advise his client to plant vine.
Frank Lloyd Right
Common forms of
Deductive Reasoning
The nature of mathematcs
Math 1
J.M.Basilla
Statements
Came from the greek word mathemata, meaning
things learned.
A mathematician wants to be sure that a certain
assertion actually follows from what has already
been accepted.
Equally as important, he wants to be sure that a
certain assertion does not follow logically from the
other.
Computations that accompany mathematics aid in
demonstrating these two points.
In some sense, mathematics is just an applied
logic.
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
The nature of mathematcs
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
In some sense, mathematics is just an applied
logic.
Logic
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
1. It is a pity that every human were given the ability to
think, but not everyone have the ability to reason
correctly.
2. Logic, the science of correct reasoning.
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Statements
Math 1
J.M.Basilla
Statements
1. By a statement, we mean a declarative sentence
which can be categorically classified as true or false.
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
2.
Argument
Statements
Non statements
Today is a holiday.
Open Sesame.
Uranus has 65 rings.
This statement is false.
All men are created equal.
What country hosted the
recent olympics?
1+2 =3
Good Luck!
3. Statements will be represented by single letters say
p,q,r, etc.
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Negation of a statement
Math 1
J.M.Basilla
The negation of a statement p is the statement
whose meaning is exactly the opposite of p.
Hence, if p is true, its negation, denoted by ∼ p, is
false.
Likewise, if p is false then ∼ p is true.
Some easy example
Statement p
Negation ∼ p
Juan is running.
Juan is not running.
Jane did not pass Math I
last semester.
Jane passed Math I last
semester.
One foot is equal to 10
inches.
One foot is not equal to 10
inches.
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Negation of statements involving quantifier
Math 1
J.M.Basilla
Statements
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.
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Example
Common forms of
Deductive Reasoning
Statement p
Negation ∼ p
All students are diligent
Some students are not
diligent
Some people are funny
No person is funny.
Nobody stole the cookie
from the cookie jar
Somebody
stole
the
cookie from the cookie jar
General rule for negating statements
involving quantifier
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Statement p
Negation ∼ p
All/Every . . ..
Some . . . not . . ..
Some . . . not . . ..
All/every . . ..
None/No/No
one/Nobody. . ..
Some . . ..
Some . . ..
No/None . . ..
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
More example and exercises
Math 1
J.M.Basilla
Statements
Statement
Negation
My car did not start.
My car did start.
Some of the cars did not
start.
All the cars started.
None of the car started.
Some of the cars did not
start.
Every car started.
Some of the cars did not
start.
Some of the cars started.
No car started.
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Conditional Statements
Math 1
J.M.Basilla
Statements
A conditional statement is usually stated in the form ”
If P then Q.”
The statement P , is called the hypothesis or
antecedent of the conditional statement.
The statement Q is called the conclusion or
consequent of the conditional statement.
In symbol, ”If P then Q.” is written as P → Q.
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.
An example:
If you study hard, then you will graduate with honors.
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
The different form of conditional statements
Math 1
J.M.Basilla
Statements
Conditional
Statements
P : You study hard.
Q : You will graduate with honors.
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
If P then Q.
If you study hard then you will graduate
with honors.
P implies
Q.
Studying hard implies that you will graduate with honors.
All P are Q.
All those who study hard graduate with
honors.
Common forms of
Deductive Reasoning
Statements related to A → B.
Converse
Inverse
Contrapositive
B→A
∼ A →∼ B
∼ B →∼ A
If it is an IBM PC then it is a computer. True
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Let
P
Q
Argument
= It is an IBM PC.
= It is a computer.
The given conditional is of the form P → Q.
Converse
If it is a computer then it is an IBM PC.
False
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
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Statements related to A → B.
Converse
Inverse
Contrapositive
B→A
∼ A →∼ B
∼ B →∼ A
If x is an even number then the last digit of x is
2. False
Let
P
Q
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
= x is even.
= The last digit of x is 2.
The given conditional is of the form P → Q.
Converse
If the last digit of x is 2 then x is even.
True
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
Common forms of
Deductive Reasoning
Statements related to A → B.
Converse
Inverse
Contrapositive
B→A
∼ A →∼ B
∼ B →∼ A
All students are diligent individuals. False
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Let
P
Q
Argument
= x is a student.
= x is diligent.
The given conditional is of the form P → Q.
Converse
All diligent individuals are students.
False
Inverse
All none students are not diligent.
False
Contrapositive
All non-diligent individuals are nonstudents. False
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Statements related to A → B.
Converse
Inverse
Contrapositive
B→A
∼ A →∼ B
∼ B →∼ A
If two lines are perpendicular then two lines form
a right angle. True
Let
P
Q
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
= Two lines are perpendicular.
= Two lines form a right angle.
The given conditional is of the form P → Q.
Converse
If two lines form a right angle then the
two lines are perpendicular True
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
then the two lines are perpendicular.
Common forms of
Deductive Reasoning
Equivalent conditional statements
Math 1
J.M.Basilla
The truth value of the conditional and the
contrapositive are always the same
The truth value of the inverse and the converse are
always the same.
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.
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Equivalent conditional statements
Math 1
J.M.Basilla
The truth value of the conditional and the
contrapositive are always the same
The truth value of the inverse and the converse are
always the same.
The converse is the contrapositive of the inverse.
conditional If x is an even number then the false
last digit of x is two.
contrapositive If the last digit of x is two then x false
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
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Biconditional statements
Math 1
J.M.Basilla
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.
false
false
false
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
false
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Biconditional statements
Math 1
J.M.Basilla
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.
true
true
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
true
true
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Biconditional statements
Math 1
J.M.Basilla
In the last two examples, the conditional, inverse,
converse and contrapositive all have the same truth
value.
These type of conditional are called biconditional.
In symbol, P ↔ Q.
Read as, P if and only if Q.
Equivalently, P and Q are equivalent.
Used in definitions.
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Inductive vs Deductive Reasoning
Math 1
J.M.Basilla
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Observed Patterns → Conclusions
Figure: Inductive reasoning
Accepted premises/truths → Conclusions
Figure: Deductive Reasoning
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Hypthetical Syllogisms
Math 1
J.M.Basilla
Given three statements, P , Q and R.
Known/accepted facts:
1. If P then Q.
2. If Q then R.
Conclusion: If P then R.
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Example:
Given 1: If you live in San Fernando then you live in
Pampanga.
Given 2: If you live in Pampanga then you speak
kapangpangan.
Conclusion: If you live in San Fernando then you
speak kapangpangan.
Hypthetical Syllogisms
Math 1
J.M.Basilla
Given three statements, P , Q and R.
Known/accepted facts:
1. If P then Q.
2. If Q then R.
Conclusion: If P then R.
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Example:
Given 1: All good chess players wears glasses.
Given 2: All who wes glasses are nerds..
Conclusion: All good chess players are nerds.
Affirming the antecedent
Math 1
J.M.Basilla
Given three statements, P and Q.
Known/accepted facts:
1. If P then Q.
2. P .
Conclusion: Q.
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Example:
Given 1: If I studied for six hours, I will pass the
exams.
Given 2: I studied for six hours.
Conclusion: I will pass the exams.
Affirming the antecedent
Math 1
J.M.Basilla
Given three statements, P and Q.
Known/accepted facts:
1. If P then Q.
2. P .
Conclusion: Q.
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Example:
Given 1: All men are mortal.
Given 2: Socrates is a man.
Conclusion: Socrates is a mortal.
Denying the consequent
Math 1
J.M.Basilla
Given three statements, P and Q.
Known/accepted facts:
1. If P then Q.
2. ∼ Q.
Conclusion: ∼ P .
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Example:
Given 1: If it rains the ground will be wet.
Given 2: The ground is not wet.
Conclusion: It did not rain.
Denying the consequent
Math 1
J.M.Basilla
Given three statements, P and Q.
Known/accepted facts:
1. If P then Q.
2. ∼ Q.
Conclusion: ∼ P .
Statements
Conditional
Statements
Statements related to a
conditional
Biconditional Statements
Argument
Inductive vs Deductive
Reasoning
Common forms of
Deductive Reasoning
Example:
Given 1: All US presidents are white.
Given 2: Obama is black.
Conclusion: Obama is not a US president.