bpa 12203 mathematics for management chapter 2

BPA 12203MATHEMATICS FOR MANAGEMENT

Chapter 2

LOGIC

Dr Chandra

Mathematics for Management: Logic

Terminology

Definition

Logic is the study of arguments.

An argument is a sequence of statements in which one in-tended as a conclusion and the others, the premises, are intendedto prove or at least provide some evidence for the conclusion.

Example

Mathematics for Management: Logic

Terminology

Definition

Logic is the study of arguments.

An argument is a sequence of statements in which one in-tended as a conclusion and the others, the premises, are intendedto prove or at least provide some evidence for the conclusion.

Example

Mathematics for Management: Logic

Example*

Mathematics for Management: Logic

Propositions (or Statement)

Definition

A proposition or statement is a declarative sentence that can beclassified as either true or false, but not both.

Note that commands, requests, questions and exclamationsare not classified as propositions.

Mathematics for Management: Logic

Example*: Which of the following are propositions? Then,determine whether they are true or false.

Mathematics for Management: Logic

Simple statement

Is a single sentence that does not contain any other statement as apart.

Lower - case letters such as p, q, r . . . as symbols for simplestatements.

Example*

p: Kuala Lumpur is the capital of Malaysia

q: Malaysia is a middle east country.

Mathematics for Management: Logic

Simple statement

Is a single sentence that does not contain any other statement as apart.

Lower - case letters such as p, q, r . . . as symbols for simplestatements.

Example*

p: Kuala Lumpur is the capital of Malaysia

q: Malaysia is a middle east country.

Mathematics for Management: Logic

Compound statement

Is a sentence with two or more simple statements as parts or whatwe call components.

Example*

Kuala Lumpur is the capital of Malaysia and Malaysia is aMiddle East country.

Mathematics for Management: Logic

Compound statement

Is a sentence with two or more simple statements as parts or whatwe call components.

Example*

Kuala Lumpur is the capital of Malaysia and Malaysia is aMiddle East country.

Mathematics for Management: Logic

Connectives*

Definition

An operator (or connective) joins simple statements into compoundstatement.

Mathematics for Management: Logic

Connectives*

Definition

An operator (or connective) joins simple statements into compoundstatement.

Mathematics for Management: Logic

Conjunction.*

A statement of the form ” p and q”.

Represented symbolically by ”p ∧ q”.

p ∧ q is true, if both p and q are true.

Otherwise, it is false.

Negation.*

A statement of the form ”not p”.

Represented symbolically by ”∼ p”.

The proposition ∼ p is true if p false and vice versa.

Mathematics for Management: Logic

Conjunction.*

A statement of the form ” p and q”.

Represented symbolically by ”p ∧ q”.

p ∧ q is true, if both p and q are true.

Otherwise, it is false.

Negation.*

A statement of the form ”not p”.

Represented symbolically by ”∼ p”.

The proposition ∼ p is true if p false and vice versa.

Mathematics for Management: Logic

Conjunction.*

A statement of the form ” p and q”.

Represented symbolically by ”p ∧ q”.

p ∧ q is true, if both p and q are true.

Otherwise, it is false.

Negation.*

A statement of the form ”not p”.

Represented symbolically by ”∼ p”.

The proposition ∼ p is true if p false and vice versa.

Mathematics for Management: Logic

Conjunction.*

A statement of the form ” p and q”.

Represented symbolically by ”p ∧ q”.

p ∧ q is true, if both p and q are true.

Otherwise, it is false.

Negation.*

A statement of the form ”not p”.

Represented symbolically by ”∼ p”.

The proposition ∼ p is true if p false and vice versa.

Mathematics for Management: Logic

Conjunction.*

A statement of the form ” p and q”.

Represented symbolically by ”p ∧ q”.

p ∧ q is true, if both p and q are true.

Otherwise, it is false.

Negation.*

A statement of the form ”not p”.

Represented symbolically by ”∼ p”.

The proposition ∼ p is true if p false and vice versa.

Mathematics for Management: Logic

Conjunction.*

A statement of the form ” p and q”.

Represented symbolically by ”p ∧ q”.

p ∧ q is true, if both p and q are true.

Otherwise, it is false.

Negation.*

A statement of the form ”not p”.

Represented symbolically by ”∼ p”.

The proposition ∼ p is true if p false and vice versa.

Mathematics for Management: Logic

Conjunction.*

A statement of the form ” p and q”.

Represented symbolically by ”p ∧ q”.

p ∧ q is true, if both p and q are true.

Otherwise, it is false.

Negation.*

A statement of the form ”not p”.

Represented symbolically by ”∼ p”.

The proposition ∼ p is true if p false and vice versa.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Inclusive disjunction. *

A statement of the form ” p or q”.

Represented symbolically by ”p ∨ q”.

p ∨ q is false, if both p and q are false.

In all other cases, it is true.

The word ”or” means ”one or the other, or both.”

Exclusive disjunction. *

A statement of the form ”either p or q”.

Represented symbolically by ”p Y q”.

p Y q is true, if exactly one of the proposition p or q is true.

The word ”either. . .or” means ”one or the other, but notboth”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.• ”p implies q and q implies p”.• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.• ”p implies q and q implies p”.• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.• ”p implies q and q implies p”.• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.• ”p implies q and q implies p”.• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.

• ”p implies q and q implies p”.• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.• ”p implies q and q implies p”.

• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.• ”p implies q and q implies p”.• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Conditional. *

A statement of the form ”if p then q” or ”p implies q”.

Represented symbolically by ”p⇒ q”.

A conditional / implication statement is false if p is true and qis false.

In other cases, a conditional statement is true.

Biconditional. *

A statement of the form

• ”p is necesssary and sufficient for q”.• ”p implies q and q implies p”.• ”p if and only if (iff) q”.

Represented symbolically by ”p⇔ q”.

Mathematics for Management: Logic

Example*

Mathematics for Management: Logic

Truth table

Definition

A truth table is a complete list of the possible truth values of astatement.

”T” for ”True” and ”F” for ”False”.

One proposition*

Mathematics for Management: Logic

Truth table

Definition

A truth table is a complete list of the possible truth values of astatement.

”T” for ”True” and ”F” for ”False”.

One proposition*

Mathematics for Management: Logic

Two propositions*

Mathematics for Management: Logic

Two propositions*

Mathematics for Management: Logic

Three propositions*

Mathematics for Management: Logic

Three propositions*

Mathematics for Management: Logic

Basic truth tables

Mathematics for Management: Logic

Example: Complete the truth table*

p q p ∧ q (p ∧ q) ∨ p ∼ q ∼ q ⇒ p (∼ q ⇒ p)⇔ [(p ∧ q) ∨ p]

T T

T F

F T

F F

Mathematics for Management: Logic

Example: Complete the truth table*

p q p ∧ q (p ∧ q) ∨ p ∼ q ∼ q ⇒ p (∼ q ⇒ p)⇔ [(p ∧ q) ∨ p]

T T T T F T T

T F F T T T T

F T F F F T F

F F F F T F T

Mathematics for Management: Logic

Tautology

If the statement is always TRUE, then this is called a tautology.

Example*: Show that the proposition (p ∧ q)⇒ p is atautology.

Mathematics for Management: Logic

Tautology

If the statement is always TRUE, then this is called a tautology.

Example*: Show that the proposition (p ∧ q)⇒ p is atautology.

Mathematics for Management: Logic

Contradiction

If the statement is always FALSE, then this is called a contradiction.

Example*: Show that the proposition (p ∧ q)∧ ∼ (p ∧ q) is acontradiction.

Mathematics for Management: Logic

Contradiction

If the statement is always FALSE, then this is called a contradiction.

Example*: Show that the proposition (p ∧ q)∧ ∼ (p ∧ q) is acontradiction.

Mathematics for Management: Logic

Logical equivalence

If two propositions p and q have the same truth values in everypossible case, the propositions are called logically equivalent.

This relationship is denoted by p ≡ q.

Example*: Show that the proposition ∼ (p ∧ q) ≡∼ p∨ ∼ q.

Mathematics for Management: Logic

Logical equivalence

If two propositions p and q have the same truth values in everypossible case, the propositions are called logically equivalent.

This relationship is denoted by p ≡ q.

Example*: Show that the proposition ∼ (p ∧ q) ≡∼ p∨ ∼ q.

Mathematics for Management: Logic

Converse *p⇒ qconverse

q ⇒ p

Contrapositive *p⇒ qcontrapositive∼ q ⇒∼ p

Inverse *p⇒ qinverse∼ p⇒∼ q

Mathematics for Management: Logic

Converse *p⇒ qconverseq ⇒ p

Contrapositive *p⇒ qcontrapositive∼ q ⇒∼ p

Inverse *p⇒ qinverse∼ p⇒∼ q

Mathematics for Management: Logic

Converse *p⇒ qconverseq ⇒ p

Contrapositive *p⇒ qcontrapositive

∼ q ⇒∼ p

Inverse *p⇒ qinverse∼ p⇒∼ q

Mathematics for Management: Logic

Converse *p⇒ qconverseq ⇒ p

Contrapositive *p⇒ qcontrapositive∼ q ⇒∼ p

Inverse *p⇒ qinverse∼ p⇒∼ q

Mathematics for Management: Logic

Converse *p⇒ qconverseq ⇒ p

Contrapositive *p⇒ qcontrapositive∼ q ⇒∼ p

Inverse *p⇒ qinverse

∼ p⇒∼ q

Mathematics for Management: Logic

Converse *p⇒ qconverseq ⇒ p

Contrapositive *p⇒ qcontrapositive∼ q ⇒∼ p

Inverse *p⇒ qinverse∼ p⇒∼ q

Mathematics for Management: Logic

Example: Consider the statements

p: You are a hardworking student.

q: You will get good marks in examination.

converse*

If you get good marks in examination, then you are ahardworking student.

contrapositive *

If you didn’t get good marks in examination, then you are nota hardworking student .

inverse *

If you are not a hardworking student, then you will not getgood marks in examination .

Mathematics for Management: Logic

Example: Consider the statements

p: You are a hardworking student.

q: You will get good marks in examination.

converse*

If you get good marks in examination, then you are ahardworking student.

contrapositive *

If you didn’t get good marks in examination, then you are nota hardworking student .

inverse *

If you are not a hardworking student, then you will not getgood marks in examination .

Mathematics for Management: Logic

Example: Consider the statements

p: You are a hardworking student.

q: You will get good marks in examination.

converse*

If you get good marks in examination, then you are ahardworking student.

contrapositive *

If you didn’t get good marks in examination, then you are nota hardworking student .

inverse *

If you are not a hardworking student, then you will not getgood marks in examination .

Mathematics for Management: Logic

Example: Consider the statements

p: You are a hardworking student.

q: You will get good marks in examination.

converse*

If you get good marks in examination, then you are ahardworking student.

contrapositive *

If you didn’t get good marks in examination, then you are nota hardworking student .

inverse *

If you are not a hardworking student, then you will not getgood marks in examination .

Mathematics for Management: Logic

Example: Consider the statements

p: You are a hardworking student.

q: You will get good marks in examination.

converse*

If you get good marks in examination, then you are ahardworking student.

contrapositive *

If you didn’t get good marks in examination, then you are nota hardworking student .

inverse *

If you are not a hardworking student, then you will not getgood marks in examination .

Mathematics for Management: Logic

Example: Consider the statements

p: You are a hardworking student.

q: You will get good marks in examination.

converse*

If you get good marks in examination, then you are ahardworking student.

contrapositive *

If you didn’t get good marks in examination, then you are nota hardworking student .

inverse *

If you are not a hardworking student, then you will not getgood marks in examination .

Mathematics for Management: Logic

Example: Consider the statements

p: You are a hardworking student.

q: You will get good marks in examination.

converse*

If you get good marks in examination, then you are ahardworking student.

contrapositive *

If you didn’t get good marks in examination, then you are nota hardworking student .

inverse *

If you are not a hardworking student, then you will not getgood marks in examination .

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T

T F

F T

F F

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q ≡ ∼ q ⇒∼ p and q ⇒ p ≡ ∼ p⇒∼ q

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q

≡ ∼ q ⇒∼ p and q ⇒ p ≡ ∼ p⇒∼ q

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q ≡

∼ q ⇒∼ p and q ⇒ p ≡ ∼ p⇒∼ q

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q ≡ ∼ q ⇒∼ p

and q ⇒ p ≡ ∼ p⇒∼ q

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q ≡ ∼ q ⇒∼ p and

q ⇒ p ≡ ∼ p⇒∼ q

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q ≡ ∼ q ⇒∼ p and q ⇒ p

≡ ∼ p⇒∼ q

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q ≡ ∼ q ⇒∼ p and q ⇒ p ≡

∼ p⇒∼ q

Mathematics for Management: Logic

Example: Truth table*

p q p⇒ q q ⇒ p ∼ p ∼ q ∼ q ⇒∼ p ∼ p⇒∼ q

T T T T F F T T

T F F T F T F T

F T T F T F T F

F F T T T T T T

Conclusion: p⇒ q ≡ ∼ q ⇒∼ p and q ⇒ p ≡ ∼ p⇒∼ q

Mathematics for Management: Logic

Validity of argument

Definition

A sequence of steps that prove or disprove a statement.

Five steps:

1. Understand the situation.

2. Denote propositions or statements as logical symbols.

3. Write all the mathematical symbols by following the sequence.

4. Construct the truth table.

5. Determine the validity of arguments.

Mathematics for Management: Logic

Validity of argument

Definition

A sequence of steps that prove or disprove a statement.

Five steps:

1. Understand the situation.

2. Denote propositions or statements as logical symbols.

3. Write all the mathematical symbols by following the sequence.

4. Construct the truth table.

5. Determine the validity of arguments.

Mathematics for Management: Logic

Validity of argument

Definition

A sequence of steps that prove or disprove a statement.

Five steps:

1. Understand the situation.

2. Denote propositions or statements as logical symbols.

3. Write all the mathematical symbols by following the sequence.

4. Construct the truth table.

5. Determine the validity of arguments.

Mathematics for Management: Logic

Validity of argument

Definition

A sequence of steps that prove or disprove a statement.

Five steps:

1. Understand the situation.

2. Denote propositions or statements as logical symbols.

3. Write all the mathematical symbols by following the sequence.

4. Construct the truth table.

5. Determine the validity of arguments.

Mathematics for Management: Logic

Validity of argument

Definition

A sequence of steps that prove or disprove a statement.

Five steps:

1. Understand the situation.

2. Denote propositions or statements as logical symbols.

3. Write all the mathematical symbols by following the sequence.

4. Construct the truth table.

5. Determine the validity of arguments.

Mathematics for Management: Logic

Validity of argument

Definition

A sequence of steps that prove or disprove a statement.

Five steps:

1. Understand the situation.

2. Denote propositions or statements as logical symbols.

3. Write all the mathematical symbols by following the sequence.

4. Construct the truth table.

5. Determine the validity of arguments.

Mathematics for Management: Logic

Example 1: Determine the validity of the argument below.

Suppose it is true that either Shalin broke the vase or Shalin waspunished, but not both and Shalin was not punished. Prove thatShalin broke the vase.

1. Understand the situation.

p: Shalin broke the vase.

q: Shalin was punished.

Mathematics for Management: Logic

Example 1: Determine the validity of the argument below.

Suppose it is true that either Shalin broke the vase or Shalin waspunished, but not both and Shalin was not punished. Prove thatShalin broke the vase.

1. Understand the situation.

p: Shalin broke the vase.

q: Shalin was punished.

Mathematics for Management: Logic

2. Denote propositions or statements as logical symbols.

Mathematics for Management: Logic

3. Write all the mathematical symbols by following thesequence.

Mathematics for Management: Logic

4. Construct the truth table.

5. Determine the validity of arguments.Since we have all TRUE (tautology), the argument is valid.

Mathematics for Management: Logic

4. Construct the truth table.

5. Determine the validity of arguments.Since we have all TRUE (tautology), the argument is valid.

Mathematics for Management: Logic

Example 2: Determine the validity of the argument below.

Suppose it is true that it is raining and if it is hot, then it is notraining and if it is not hot, then I cannot go fishing. Prove that Icannot go fishing.

1. Understand the situation.

p: It is raining.

q: It is hot.

r: I can go fishing.

Mathematics for Management: Logic

Example 2: Determine the validity of the argument below.

Suppose it is true that it is raining and if it is hot, then it is notraining and if it is not hot, then I cannot go fishing. Prove that Icannot go fishing.

1. Understand the situation.

p: It is raining.

q: It is hot.

r: I can go fishing.

Mathematics for Management: Logic

2. Denote propositions or statements as logical symbols.

Mathematics for Management: Logic

3. Write all the mathematical symbols by following thesequence.

Mathematics for Management: Logic

4. Construct the truth table.

5. Determine the validity of arguments.Since we have all TRUE (tautology), the argument is valid.

Mathematics for Management: Logic

4. Construct the truth table.

5. Determine the validity of arguments.Since we have all TRUE (tautology), the argument is valid.

Mathematics for Management: Logic

Example 3: Determine the validity of the argument below.

If I have finished my meal, then I will go to the book store.If I go to the book store, then my brother will also go to the book store.My brother did not go to the book store, so I haven’t finished mymeal.

1. Understand the situation.

p: I have finished my meal.

q: I will go to the book store.

r: My brother will also go to the book store.

Mathematics for Management: Logic

Example 3: Determine the validity of the argument below.

If I have finished my meal, then I will go to the book store.If I go to the book store, then my brother will also go to the book store.My brother did not go to the book store, so I haven’t finished mymeal.

1. Understand the situation.

p: I have finished my meal.

q: I will go to the book store.

r: My brother will also go to the book store.

Mathematics for Management: Logic

2. Denote propositions or statements as logical symbols.

Mathematics for Management: Logic

3. Write all the mathematical symbols by following thesequence.

Mathematics for Management: Logic

4. Construct the truth table.

5. Determine the validity of arguments.Since we have all TRUE (tautology), the argument is valid.

Mathematics for Management: Logic

4. Construct the truth table.

5. Determine the validity of arguments.Since we have all TRUE (tautology), the argument is valid.

Mathematics for Management: Logic