CS113-Week1

Logic Week 1 - Garima Mandal

Page 1: Title

Logic Week 1 by Garima Mandal

Page 2: Agenda

Overview Topics Include:

• Statements: Understanding the basic building blocks of logical reasoning.

• Negation: Exploring how to create oppositional statements.

• Conjunction and Disjunction: Analyzing how statements can be combined.

• Implications: Delving into conditional statements and their nuances.

• Biconditionals: Understanding statements that are true in both directions.

• Tautologies and Contradictions: Identifying statements that are always true and always false.

• Rules of Inference: Learning the principles governing valid logical arguments.

• Applications of Logic: Considering practical uses of logic in various fields such as mathematics, computer science, and philosophy.

Page 3: Statements Introduction

Definition:Discrete mathematics focuses on objects with distinct separated values, crucial for working with logical statements.Statements are the fundamental building blocks of logical reasoning and proofs in discrete mathematics.

Importance of Statements:

• Understanding: Grasping the nature of statements is essential for comprehending mathematical concepts and frameworks.

• Usage: Statements form the basis for constructing logical arguments and proofs; they support complex mathematical ideas.

• Truth: The validity of reasoning hinges on the truth of foundational statements; false statements lead to incorrect conclusions and flawed logic.

Page 4: Verifying Statements

Accepting Truth of Statements:Among statements, some, known as axioms, are accepted as true without requiring proof; they are considered fundamental principles within mathematical systems.

Showing Statements to be True:Most other statements must be shown to be true through rigorous proof, employing logical reasoning and established methods.

Verification Methods:Clear and accepted verification methods are critical in mathematics; peer review, reproducibility, and the establishment of a logical foundation matter significantly.Developing skills for self-verification enhances mathematical thinking and involves mastering various proof techniques.

Page 5: Understanding Statements

Definition of a Statement:A statement is defined as a declarative sentence that can unequivocally be categorized as either true or false.

Types of English Sentences:

• Declarative: Asserts a fact (e.g., "The sum of angles in a triangle is 180 degrees.")

• Interrogative: Asks a question (e.g., "What is the value of x in the equation 2x + 5 = 15?")

• Imperative: Gives a command (e.g., "Solve for x in the equation 2x + 5 = 15.")

• Exclamatory: Expresses strong emotion (e.g., "Wow! The solution was so elegant!")

Page 6: Statements in Discrete Mathematics

The focus is primarily on declarative sentences that possess a clear truth value - they are either true or false.Truth Value:Each mathematical statement must have a clear truth value, allowing for the evaluation of its truth.

Conclusion:Understanding statements is foundational for logical reasoning and is essential for advanced studies in mathematics and related fields like computer science, philosophy, and logic.

Page 7: What Constitutes a Statement

Statement Definition:A statement is a definite declarative sentence assessable as either true or false, which is specific and avoids use of undefined variables.

Examples of Statements:

• "The Earth orbits the Sun." (True)

• "2 + 2 = 5" (False)

• "Water boils at 100°C at sea level." (True)

Not Statements:Questions, commands, exclamations, and sentences with undefined variables (e.g., "Let x be a number.") cannot qualify as statements due to their lack of a definitive truth value.

Page 8: Truth Values

Context Impact on Truth Values:For instance, the statement "Donald Trump is the President" was True in 2019 but is False in 2023, highlighting the importance of temporal context.Truth values are central to logic and are denoted as T (True) or F (False).

Classical Logic Principle:According to the law of excluded middle, a statement must be either true or false, with no middle ground.

Mathematical Examples under Truth Values:Statements such as "The square root of 2 is irrational" have definitive truth values that can be substantiated through proof.

Page 9: Continued Explanation of Truth Values

Logical Arguments:Mathematical proofs employ logical arguments for the verification of statements, ensuring rigorous integrity in reasoning.

Open Sentences:Defined as declarative sentences containing variables, their truth value is contingent on the assigned values to those variables.

Page 10: Negation, Conjunction, and Disjunction

An inquiry into logical statement relationships and combinations leads to the formation of new statements.Truth Table Purpose:A vital tool for analyzing the truth values of combined statements, relying heavily on the definitions of logical operations.

Single Statement Truth Values:There are only two possibilities for truth values: True (T) or False (F).

Combining Two Statements:When two statements (P and Q) are combined, four possible combinations of truth values arise, illustrating all truth possibilities.

Page 11: Importance of Truth Tables

Truth tables effectively display relationships among statements, support logical operations, and provide clarity in reasoning concerning operators like AND, OR, and NOT.

Page 12: Negation

Definition:The negation of a statement P is expressed as not P, shown symbolically as ∼ P.The negation creates a new statement that represents the opposite of the original statement.

Page 13: Conjunction

Definition & Symbol:Conjunction asserts that both statements P and Q are true, denoted as P ∧ Q.Truth Table for Conjunction:A truth table reveals that P ∧ Q is true only when both statements P and Q are true, ensuring both conditions are satisfied for the conjunction to hold.

Page 14: Disjunction

Definition:Disjunction is expressed as P ∨ Q, being true when at least one of the statements is true.Interpretation Variants:

• Inclusive OR: True when at least one statement is true, potentially both.

• Exclusive OR: True if exactly one statement is true, but not both, leading to distinct logical interpretations.

Page 15: Inclusive and Exclusive Or

Distinctions:The Inclusive OR (P ∨ Q) allows for both statements to be true concurrently, whereas Exclusive OR (P ⊕ Q) allows for only one to be true.

Page 16: Truth Tables for XOR and OR

Representation:Truth tables illustrate the truth values and highlight the distinctions between inclusive and exclusive OR operations clearly.

Page 17: Compound Statements

Logical Connectives Defined:Negation (∼), Conjunction (∧), Disjunction (∨), Exclusive OR (⊕).Truth Values:Determined by the component statements and the logical connectives applied, forming the basis for evaluating compound statements.

Page 18: Logical Equivalence

Definition:Two compound statements are considered logically equivalent if they share truth values across all combinations regardless of the specific values assigned.Examples of Equivalence:

• P ∧ Q ≡ Q ∧ P (Commutative property)

• P ∨ Q ≡ Q ∨ P (Also Commutative)

Page 19: De Morgan's Laws

Theorems:For statements P and Q, De Morgan's Laws articulate the interactions between negation and logical connectors:

• ~(P ∨ Q) ≡ (~P) ∧ (~Q)

• ~(P ∧ Q) ≡ (~P) ∨ (~Q)

Page 20: Associative and Distributive Laws

Associative Properties in Logic:Similar to arithmetic, these properties are applicable for both conjunction and disjunction operations.

Distributive Properties:There are two distinct distributive laws in logic, governing operations that interplay between statements, ensuring logical consistency.

Page 21: Applications of Laws

Practical Applications:These laws are instrumental in simplifying logical expressions, which find applications across programming, artificial intelligence, and mathematical proofs, enhancing the efficiency of logical reasoning.

Page 22: Implications

Definition:A conditional statement represented as P ⇒ Q illustrates a logical relationship where the truth of P entails the truth of Q.Truth Value Analysis:A clear understanding of the truth table dictates the validity of implications, emphasizing the necessary conditions for their truth.

Page 23: Implication Interpretation

The concept of vacuous truth within implications emphasizes statements that are technically true due to the failure of their conditions, aiding in deeper logical reasoning.

Page 24: Stating Implications in Words

Different verbal expressions can convey the same underlying implications, illustrating their equivalences and nuances in logical discourse.

Page 25: Converse of an Implication

Definition:The converse of an implication is given by Q ⇒ P, emphasizing its difference from the original P ⇒ Q and the implications therein.

Page 26: Contrapositive of an Implication

This consists of negating the statements involved and reversing their positions, critical for constructing proofs within mathematics and ensuring logical integrity.

Page 27: Logical Equivalence Verified

Truth tables effectively demonstrate the logical equivalence between an implication and its contrapositive, reinforcing the relationships among logical statements.

Page 28: Q&A Session

Closing remarks and an open invitation for questions to enhance understanding and clarify concepts introduced throughout the week.