Ma 124, Sem I, 2011-2012 Logic

Logic

Mark Tolentino

Mathematics DepartmentSchool of Science and Engineering

Ateneo de Manila University

15 June – 08 July 2011

Ma 124, Sem I, 2011-2012 Logic

Logic Puzzles

Ma 124, Sem I, 2011-2012 Logic

There are three boxes. One is labeled "APPLES" another is labeled "ORANGES". The last one is labeled "APPLES AND ORANGES". You know that each is labeled incorrectly. You may ask me to pick one fruit from one box which you choose.

How can you label the boxes correctly?

Puzzle 1

Ma 124, Sem I, 2011-2012 Logic

Puzzle2

1. The first question with B as the correct answer is:A. 1B. 4C. 3D. 2

3. The correct answer to Question 1 is:

A. DB. CC. BD. A

2. The correct answer to Question 4 is:

A. DB. AC. BD. C

4. The number of questions which have D as the correct answer is:A. 3B. 2C. 1D. 0

5. The number of questions which have B answer is:

A. 0B. 2

C. 3D. 1

Ma 124, Sem I, 2011-2012 Logic

Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a “Yes” or “No” response to a question a tourist asks. Suppose you are a tourist visiting this area. You come to a fork in the road. One branch leads to the ruins you want to visit, the other branch leads deep into the jungle. A villager is standing at the fork in the road. What one question can you ask the villager to determine which branch to take?

Puzzle 3

• Logical reasoning is indispensable in discussions and/or arguments in any field of study, research, or work.

• The rules of logic give precise meaning to mathematical statements. These rules are used to distinguish between valid and invalid mathematical arguments.

• Logic has numerous applications in computer science such as design of computer circuits and construction of computer programs.

• Studying logic, like studying math itself, sharpens one’s mind that enables creative and critical thinking.

Why Study Logic?

Ma 124, Sem I, 2011-2012 Logic

Ma 124, Sem I, 2011-2012 Logic

Section 1.1

Propositional Logic

A proposition is a statement that is either true or false.

Definition

Ma 124, Sem I, 2011-2012 Logic

Ma 124, Sem I, 2011-2012 Logic

Determine which of the following are propositions.a. PNoy is the president elect of the Philippines.b. What time is it?c. Take out one-half sheet of paper for a quiz.d. There are classes on June 24, 2011.e. 90 = 0.f. ex = 9.g. x2 + y2 = z2.

h. a

Example 1

sin( )If , then .

x xx

23 2 1 03æ ö- + + ÷ç ÷çÎ ³÷ç ÷÷çè ø

Ma 124, Sem I, 2011-2012 Logic

• Letters are used to represent propositions. Conventional letters are p, q, r, s, …

• The truth value of a proposition is denoted by T, if it is true, or F, if it is false.

• Compound propositions are formed from existing propositions using logical operators.

Remarks

Ma 124, Sem I, 2011-2012 Logic

Let p be a proposition. The statement“It is not the case that p.”

is another proposition, called the negation of p. The negation of p is denoted by ¬p, read “not p.”

Definition

• To study all the possible values of the negation of a proposition, a truth table is used.

Truth Table

p ¬p

T F

F T

Ma 124, Sem I, 2011-2012 Logic

Let p be the proposition “There is a quiz on Wednesday.” Write the negation of p.

Example 2

Ma 124, Sem I, 2011-2012 Logic

Let p and q be propositions. The proposition “p and q,” denoted by p ∧ q, is the proposition that is true when both p and q are true and is false otherwise. The proposition p ∧ q is called the conjunction of p and q.

Definition

Truth Table

p q p ∧ qT T TT F FF T FF F F

Ma 124, Sem I, 2011-2012 Logic

Let p be the proposition “Today is Monday” and q be the proposition “It is scorching hot today.” Write the conjunction of p and q.

Example 3

Ma 124, Sem I, 2011-2012 Logic

Let p and q be propositions. The proposition “p or q,” denoted by p ∨ q, is the proposition that is false when both p and q are false and is true otherwise. The proposition p ∨ q is called the disjunction of p and q.

Definition

Truth Table

p q p ∨ qT T TT F TF T TF F F

Ma 124, Sem I, 2011-2012 Logic

Let p be the proposition “Today is Monday” and q be the proposition “It is scorching hot today.” Write the disjunction of p and q.

Example 4

Ma 124, Sem I, 2011-2012 Logic

The statement in a fastfood selection says that “The meal comes with a slice of cake or a scoop of ice cream.” Is the “or” in this statement interpreted using definition 3?

Example 5

Ma 124, Sem I, 2011-2012 Logic

Let p and q be propositions. The exclusive or of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true and is false otherwise.

Definition

Truth Table

p q p ⊕ qT T FT F TF T TF F F

Ma 124, Sem I, 2011-2012 Logic

Let p and q be propositions. The implication p → q is the proposition that is false when p is true and q is false and is true otherwise. Here, p is called the hypothesis (or premise) and q is called the conclusion.

Definition

Truth Table

p q p → q T T TT F FF T TF F T

Ma 124, Sem I, 2011-2012 Logic

Let p be the proposition “I get elected president” and q be the proposition “The Philippines will become a first-world country.” Write the implication p → q.

Example 6

Ma 124, Sem I, 2011-2012 Logic

True or false?a. “If today is Friday, then 2 + 3 = 5.”b. “If today is Friday, then 2 + 3 = 6.”

Example 7

Ma 124, Sem I, 2011-2012 Logic

Other common ways to express p → q are• if p, then q• p implies q• if p, q• q if p• q whenever p• p is sufficient for q• q is necessary for p• p only if q

Remark

Ma 124, Sem I, 2011-2012 Logic

Write the statement “You can enter the school only if you have an ID” in symbol form. Let p be “You can enter the school” and q be “You have an ID.”

Example 8

Ma 124, Sem I, 2011-2012 Logic

• The proposition q → p is the converse of p → q.• The proposition ¬q → ¬p is the contrapositive of p → q.

Definition

Ma 124, Sem I, 2011-2012 Logic

Consider the implication “If today is a Monday, then I have a test today.”• The converse of the statement is “If I have a test today,

then today is a Monday.”• The contrapositive of the statement is “If I do not have a

test today, then today is not a Monday.”

Example 9

Ma 124, Sem I, 2011-2012 Logic

Let p and q be propositions. The biconditional p ⟷ q is the proposition that is true when p and q have the same truth values and is false otherwise.

Definition

Truth Table

p q p ⟷ qT T TT F FF T FF F T

Ma 124, Sem I, 2011-2012 Logic

• The biconditional p ⟷ q is true precisely when both p → q and q → p are true. Because of this, p ⟷ q is read as “p if and only if q.”

• Other common ways to express p ⟷ q are• p is necessary and sufficient for q• if p then q, and conversely

Remarks

Ma 124, Sem I, 2011-2012 Logic

Write the following statements in symbols.a. “You cannot ride the roller coaster if you are under 4 feet

tall unless you are older than 16 years old.”b. “You can access the Internet from campus only if you are

a computer science major or you are not a freshman.”

Example 10

Ma 124, Sem I, 2011-2012 Logic

Use truth tables to determine the possible truth values of the following propositions.a. (p → q) (∨ ¬p → q)b. (q ∧¬r) → ¬pc. p → (q ∨¬r)d. (p → q) (∧ ¬p → r)

Example 11

Ma 124, Sem I, 2011-2012 Logic

Section 1.2

Logical Equivalences

Ma 124, Sem I, 2011-2012 Logic

A compound proposition that is always true, no matter what the truth values of the propositions that occur in it, is called a tautology. A compound proposition that is always false is called a contradiction. A compound proposition that is neither a tautology nor a contradiction is called a contingency.

Definition

Ma 124, Sem I, 2011-2012 Logic

Show thata. p ∨¬p is a tautology.b. p ∧¬p is a contradiction.

Example 1

Ma 124, Sem I, 2011-2012 Logic

The compound propositions p and q are called logically equivalent if p ⟷ q is a tautology. The notation p ≡ q (or p

⟺ q) denotes that p and q are logically equivalent.

Definition

Ma 124, Sem I, 2011-2012 Logic

• ≡ or is not a logical connective but rather is the ⟺statement “p ⟷ q is a tautology.”

• The compound propositions p and q are equivalent if and only if the columns giving their truth values (in a truth table) agree.

Remarks

Ma 124, Sem I, 2011-2012 Logic

Determine if the following propositions are logically equivalent.a. p → q ; ¬q → ¬p (contrapositive)b. p → q ; q → p (converse)

Example 2

Ma 124, Sem I, 2011-2012 Logic

Common Logical Equivalences

Ma 124, Sem I, 2011-2012 Logic

Common Logical Equivalences

Ma 124, Sem I, 2011-2012 Logic

Without using a truth table, show that¬(p ∨ (¬p ∧ q))and ¬p ∧ ¬qare logically equivalent.

Example 3

Ma 124, Sem I, 2011-2012 Logic

Without using a truth table, show that(p ∧ q) → (p ∨ q)is a tautology.Example 4

Ma 124, Sem I, 2011-2012 Logic

Without using a truth table, show that¬p → (q → r)andq → (p ∨ r)

are logically equivalent.

Example 5

Ma 124, Sem I, 2011-2012 Logic

Without using a truth table, show that¬(p ⟷ q)and ¬p ⟷ qare logically equivalent.

Example 6

Ma 124, Sem I, 2011-2012 Logic

Section 1.3

Predicates and Quantifiers

Ma 124, Sem I, 2011-2012 Logic

There are statements that involve variables such as"x > 0" "x + 1 = 3" "x2 + y2 = z2"

“x loves the Philippines.”These statements are not propositions unless all variables are assigned particular values.

Statements with variables

Ma 124, Sem I, 2011-2012 Logic

Statements involving variable/s can be divided into two parts. The first part is the set of variables, which is called the subject. The second part of the statement is the predicate, a certain property or characteristic that the subject can have.

Definition

Ma 124, Sem I, 2011-2012 Logic

• In the statement “x > 0,” the subject is x and the predicate is “>0” or the property of “being positive.”

• In the statement “x loves the Philippines,” x is the subject while “loves [being loving of] the Philippines” is the predicate.

• There can also be multiple subjects as in the case of x2 + y2

= z2. Here, x, y and z are the subjects while the predicate is the property that x2 + y2 = z2.

Example 1

Ma 124, Sem I, 2011-2012 Logic

A statement that involves a variable, say x, can be represented as P(x) where P is called a propositional function. P(x), the function value of P at x, can be interpreted as the statement obtained by replacing the variable with a particular value x.

Definition

Ma 124, Sem I, 2011-2012 Logic

Let P(x) be the statement “x > 0.”• P(4) = “4 > 0” which is T.• Similarly, P(–3) = “–3 > 0” which is F.

Example 2

Ma 124, Sem I, 2011-2012 Logic

• The function values of a propositional function are propositions.

• A propositional function can involve more than one variable. In this case, the propositional function is written as P(x1, x2, …, xn) where n is the number of variables involved.

• For example, the statement “x2 + y2 = z2” can be denoted by F(x, y, z). In this case, F(3, 4, 5) = “32 + 42 = 52” which has truth value T.

Remarks

Ma 124, Sem I, 2011-2012 Logic

The domain of discourse or the universe of discourse (or simply the domain) is the set of all possible values of the variable x in the propositional function P(x).

Definition

Ma 124, Sem I, 2011-2012 Logic

A quantifier specifies the set of values of x to be considered for a propositional function P(x). Common quantifiers are all, some, and none.

Definition

Ma 124, Sem I, 2011-2012 Logic

The propositional function P(x) = “x2 ≥ 0” can be quantified using the quantifier “for all real numbers .” Using this, the statement becomes a true proposition.

Example 3

Ma 124, Sem I, 2011-2012 Logic

• Quantification is a way of making a proposition from a propositional function.

• As such, a quantification of a propositional function is a proposition; thereby, it has a truth value.

Remarks

Ma 124, Sem I, 2011-2012 Logic

The universal quantification of P(x) is the statement“P(x) for all values of x in the domain.”

The notation ∀x P(x) denotes the universal quantification of P(x). Here is called the ∀ universal quantifier and is read as “for all,” “for every,” or “for each.”

Definition

Ma 124, Sem I, 2011-2012 Logic

• ∀x P(x) is• true when P(x) is true for every x.• false when there is an x for which P(x) is false.

• An element for which P(x) is false is called a counterexample of ∀x P(x).

• When the domain is empty, ∀x P(x) is true for any propositional function P(x) because there are no elements x in the domain for which P(x) is false.

Remarks

Ma 124, Sem I, 2011-2012 Logic

Let P(x) be the statement “|sin x| ≤ 1.” What is the truth value of the quantification ∀x P(x) where the domain is all real numbers?

Example 4

Ma 124, Sem I, 2011-2012 Logic

Let P(a) be the statement

What is the truth value of the quantification ∀a P(a) where the domain is all real numbers?

Example 5

limx a x a

1 1®

æö÷ç ÷=ç ÷ç ÷çè ø

Ma 124, Sem I, 2011-2012 Logic

• Suppose that the domain of a propositional function P(x) is finite with n elements. Then, the elements of the domain can be listed, say, as x1, x2, …, xn.

• In this case, the quantification ∀x P(x) is the same as the conjunction

P(x1) ∧ P(x2) ∧ ∧⋯ P(xn).

Remarks

Ma 124, Sem I, 2011-2012 Logic

What is the truth value of ∀x P(x), where P(x) is the statement “x2 < 10” and the domain consists of positive integers not exceeding 4?

Example 6

Ma 124, Sem I, 2011-2012 Logic

The existential quantification of P(x) is the statement“There exists an element x in the domain such that P(x).”

The notation ∃x P(x) denotes the existential quantification of P(x). Here is called the ∃ existential quantifier and is read as “there exists” or “for some.”

Definition

Ma 124, Sem I, 2011-2012 Logic

• ∃x P(x) is• true when there is an x for which P(x) is true.• false when P(x) is false for every x.

• When the domain is empty, ∃x P(x) is false for any propositional function P(x) because there can be no element in the domain for which P(x) is true.

Remarks

Ma 124, Sem I, 2011-2012 Logic

Let P(x) be the statement “x2 + 4x + 4 = 0.” What is the truth value of the quantification ∃x P(x) where the domain is all real numbers?

Example 7

Ma 124, Sem I, 2011-2012 Logic

Let P(a) be the statement “|a| = –2/3.” What is the truth value of the quantification ∃a P(a) where the domain is all real numbers?

Example 8

Ma 124, Sem I, 2011-2012 Logic

• Suppose that the domain of a propositional function P(x) is finite with n elements. Then, the elements of the domain can be listed, say, as x1, x2, …, xn.

• In this case, the quantification ∃x P(x) is the same as the disjunction

P(x1) ∨ P(x2) ∨ ∨⋯ P(xn)

Remarks

Ma 124, Sem I, 2011-2012 Logic

What is the truth value of ∃x P(x), where P(x) is the statement “x2 > 10” and the domain consists of positive integers not exceeding 4?

Example 9

Ma 124, Sem I, 2011-2012 Logic

• There is no limit to the number of quantifiers that can be defined. Some examples are “there are at least 100,” “there are at most two,” and “there are exactly one million.”

• Another important quantifier is the uniqueness quantifier, denoted by ! or ∃ ∃1. The notation !∃ x P(x) or ∃1x P(x) states that “There exists a unique x such that P(x) is true.” The uniqueness quantification can also be phrased as “there is exactly one” or “there is one and only one.”

• An example of a unique quantification is !∃ x (x + 1) = 2. This is a true proposition.

Other Quantifiers

Ma 124, Sem I, 2011-2012 Logic

An abbreviated notation is often used to restrict the domain of quantifiers.

Quantifiers with Restricted Domain

a. ∀t < 0 (|t| = –t) b. ∃z > 0 (z2 > 0)

Example 10

Ma 124, Sem I, 2011-2012 Logic

• When a quantifier is used on a variable , is said to be bound.

• When a variable is not bound by a quantifier or is not assigned a particular value, is said to be free.

• The part of a logical expression to which a quantifier is applied is called the scope of this quantifier.

Other Terms

Ma 124, Sem I, 2011-2012 Logic

Statements involving predicates and quantifiers are logically equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. The notation S ≡ T is used to indicate that two statements (involving predicates and quantifiers) S and T are logically equivalent.

Definition

Ma 124, Sem I, 2011-2012 Logic

Show that ∀x (P(x) ∧ Q(x)) and ∀x P(x) ∧ ∀x Q(x) are logically equivalent (where the same domain is used throughout).

Example 11

Ma 124, Sem I, 2011-2012 Logic

Show that ∃x (P(x) ∨ Q(x)) and ∃x P(x) ∨ ∃x Q(x) are logically equivalent (where the same domain is used throughout).

Example 12

Ma 124, Sem I, 2011-2012 Logic

Negating Quantified Expressions

Ma 124, Sem I, 2011-2012 Logic

Negate the statement “Everybody student in this class loves video games.”

Example 13

Ma 124, Sem I, 2011-2012 Logic

Negate the statement “There is a student in this class who loves math.”

Example 14

Ma 124, Sem I, 2011-2012 Logic

Section 1.4

Nested Quantifiers

Ma 124, Sem I, 2011-2012 Logic

Nested Quantifiers

Ma 124, Sem I, 2011-2012 Logic

Let Q(x, y) denote the statement “x + y = 0.” Let the domain of both x and y be ℝ. What is the truth value of each of the following?a. ∀x∀y Q(x, y)b. ∃x∃y Q(x, y)c. ∀x∃y Q(x, y)d. ∃x∀y Q(x, y)

Example 1

Ma 124, Sem I, 2011-2012 Logic

Let Q(x, y, z) denote the statement “x + y = z.” Let the domain of all variables be ℝ. What is the truth value of each of the following?a. ∀x∀y∃z Q(x, y, z)b. ∃z∀x∀y Q(x, y, z)c. ∃x∀y∃z Q(x, y, z)

Example 2

Ma 124, Sem I, 2011-2012 Logic

Which proposition is true?∀x∃y ((x ≠ 0) → (xy = 1))

or∃y∀x ((x ≠ 0) → (xy = 1))

Example 3

Ma 124, Sem I, 2011-2012 Logic

Let F(x, y) = “x and y are friends.” Translate the following statement to English.

∃x∀y∀z ((F(x, y) ∧ F(x, z) (∧ y ≠ z)) → ¬F(y, z))

Example 4

Ma 124, Sem I, 2011-2012 Logic

Write the statement “Everyone has exactly one best friend” using predicates and quantifiers where the domain of discourse is all human beings.

Example 5

Ma 124, Sem I, 2011-2012 Logic

Recall the definition of limit from calculus.(a) Write the definition of limit using predicates and

quantifiers. Let ℝ be the domain of all variables involved.(b) Use predicates and quantifiers to express limit of f(x) as x

approaches a is not equal to L.(c) Use predicates and quantifiers to express that limit of

f(x) as x approaches a does not exist.

Example 6

Ma 124, Sem I, 2011-2012 Logic

Section 1.5

Rules of Inference

Ma 124, Sem I, 2011-2012 Logic

The semester has just started, rumors are going around that “Our math teacher only gives an F, a C, or an A as final grade.”

Motivation

Ma 124, Sem I, 2011-2012 Logic

As the math teacher explains the syllabus, he says that “A student will not get an F if he passes all the exams.”

Motivation

Ma 124, Sem I, 2011-2012 Logic

Monica, scared of getting an F, studied hard for the entire semester. Before the release of grades, her math teacher tells her that “You will not get an F.”

Motivation

Ma 124, Sem I, 2011-2012 Logic

Did Monica pass all the exams?

Question 1

Ma 124, Sem I, 2011-2012 Logic

Monica knew that she would really not get an F because long test scores are B+, B, A, B. She wanted to know her final grade so she asked the math teacher.

Motivation

Ma 124, Sem I, 2011-2012 Logic

The math teacher tells her “You will not get an A if at least two of your exam scores are at most a B.”

Motivation

Ma 124, Sem I, 2011-2012 Logic

What is Monica’s final grade?

Question 2

Ma 124, Sem I, 2011-2012 Logic

An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. An argument form in propositional logic is a sequence of compound propositions involving propositional variables. An argument form is valid if no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true.

Definition

Ma 124, Sem I, 2011-2012 Logic

An argument form with premises p1, p2, …, pn and conclusion q is valid when (p1 ∧ p2 ∧ ∧⋯ pn) → q is a tautology.

Remark

Ma 124, Sem I, 2011-2012 Logic

Use the following true statements• “If it does not rain or if it is not foggy, then the sailing race

will be held and the lifesaving demonstration will go on." • "If the sailing race is held, then the trophy will be

awarded."• "The trophy was not awarded." to conclude that “It rained.”

Example 1

Ma 124, Sem I, 2011-2012 Logic

Rules of Inference

Ma 124, Sem I, 2011-2012 Logic

Rules of Inference

Ma 124, Sem I, 2011-2012 Logic

Letr “It rained.”f “It is foggy.”s “The sailing race is held.”l “The lifesaving demonstration goes on.”t “The trophy is awarded.” Use rules of inference to show that the following argument/argument form is valid. 

Example 2

(¬r ∨¬f) → (s ∧ l)

s → t ¬t

∴ r

Ma 124, Sem I, 2011-2012 Logic

Use rules of inference to show that the following argument/argument form is valid. 

Example 3

(p ∧ t) → (r ∨ s)

q → (u ∧ t)

u → p¬sq

∴ r

Ma 124, Sem I, 2011-2012 Logic

Rules of Inference for Quantifies Statements

Ma 124, Sem I, 2011-2012 Logic

Show that the following argument is valid by showing that its argument form is valid using rules of inference. 

Example 4

“Everyone in this class has taken a course in calculus.”

“John is in this class.”

“∴ John has taken calculus.”

Ma 124, Sem I, 2011-2012 Logic

Show that the following argument is valid by showing that its argument form is valid using rules of inference. 

Example 5

“Every student can access the Internet.”

“Everyone has a laptop or can access the Internet.”

”∴ Someone who is not a student has a laptop.”

Ma 124, Sem I, 2011-2012 Logic

• Steps 5, 6 and 7 are common steps in logical reasoning. That is why a combined rule called universal modus tollens is defined as follows:

 

• The similar rule called universal modus ponens is defined as follows: 

Remarks

∀x (P(x) → Q(x)) ¬Q(a) where a is a particular element in domain

∴ ¬P(a)

∀x (P(x) → Q(x))

P(a) where a is a particular element in domain

∴ Q(a)

Ma 124, Sem I, 2011-2012 Logic

Show that the following argument form is valid using rules of inference. 

Example 6

∀x (P(x) → (Q(x) ∧ S(x)))

∀x (P(x) ∧ R(x))

∴ ∀x (R(x) ∧ S(x))

Ma 124, Sem I, 2011-2012 Logic

Show that the following argument form is valid using rules of inference.

Example 7

∀x (P(x) ∨ Q(x))

∀x (¬Q(x) ∨ S(x))

∀x (R(x) → ¬S(x) )

∃x ¬P(x)

∴ ∃x ¬R(x)