Logic J.M.Basilla

Introduction to Logic Math 1 General Mathematics Lecture 7 Julius Magalona Basilla Institute of Mathematics University of the Philippines-Diliman [email protected]

2011 Math 1

Statements Propositional Logic Negation Conditional

Some quotes Logic J.M.Basilla Statements Propositional Logic

A doctor can bury his mistakes but an architect can only advise his client to plant vine. Frank Lloyd Right When a mathematican makes a mistake, he simply erases the board.

Negation Conditional

The nature of mathematcs Logic J.M.Basilla

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.

Statements Propositional Logic Negation Conditional

The nature of mathematcs Logic J.M.Basilla Statements Propositional Logic Negation Conditional

In some sense, mathematics is just an applied logic.

Logic 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. 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.

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

Logic Logic J.M.Basilla Statements Propositional Logic Negation Conditional

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.

Statements Logic

1. By a statement, we mean a declarative sentence which can be categorically classified as true or false.

J.M.Basilla Statements Propositional Logic

2.

Negation

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.

Conditional

Some english sentences which are not a mathematical statements A caveat! Logic J.M.Basilla Statements Propositional Logic Negation

Not all english sentences are mathematical statement. ” I am lying now.” is not a mathematical statement. ”This statement is false.” is also not a mathematical statement.

Conditional

Negation of a statement Logic

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.

J.M.Basilla Statements Propositional Logic Negation Conditional

Negation of statements involving quantifier Logic J.M.Basilla

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.

Statements Propositional Logic Negation Conditional

Example 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 Logic J.M.Basilla Statements

Statement p

Negation ∼ p

Propositional Logic Negation Conditional

All/Every . . ..

Some . . . not . . ..

Some . . . not . . ..

All/every . . ..

None/No/No one/Nobody. . ..

Some . . ..

Some . . ..

No/None . . ..

More example and exercises Logic J.M.Basilla

Statement

Negation

My car did not start.

My car did start.

Statements Propositional Logic Negation Conditional

Some of the cars did not start.

All the cars started.

None of the car started.

Some of the cars did start.

Every car started.

Some of the cars did not start.

Some of the cars started.

No car started.

The truth of a negation p T F

∼p F T

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

Conditional Statements Logic J.M.Basilla

A conditional statement is usually stated in the form ” If P then Q.”

Statements Propositional Logic

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.

Negation Conditional

The truth of a conditional p T T F F

q T F T F

p =⇒ q T F T T

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

The different form of conditional statements Logic J.M.Basilla

P : You study hard. Q : You will graduate with honors.

Statements Propositional Logic Negation Conditional

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.

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

Logic J.M.Basilla Statements Propositional Logic Negation

Let P Q

Conditional

= 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

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

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

Let P Q

= 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

Statements related to A → B. Converse Inverse Contrapositive

B→A ∼ A →∼ B ∼ B →∼ A

All students are diligent individuals. False

Logic J.M.Basilla Statements Propositional Logic Negation

Let P Q

Conditional

= 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

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

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

Let P Q

= 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

Equivalent conditional statements 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.

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

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.

Equivalent conditional statements 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.

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

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

Biconditional statements 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

Logic J.M.Basilla Statements

false

Propositional Logic Negation Conditional

false

Biconditional statements 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.

Logic

true

J.M.Basilla

true

Statements Propositional Logic Negation Conditional

true

true

Biconditional statements In the last two examples, the conditional, inverse, converse and contrapositive all have the same truth value. These type of conditional are called biconditional.

Logic J.M.Basilla Statements Propositional Logic Negation Conditional

In symbol, P ↔ Q. Read as, P if and only if Q. Equivalently, P and Q are equivalent. Used in definitions.

The truth of a biconditional p T T F F

q T F T F

p ⇐⇒ q T F F T

Logic J.M.Basilla Statements Propositional Logic Negation Conditional