Module 2 Part 3

Module Overview

Title: Mathematical Language & SymbolsCourse: Mathematics in the Modern WorldInstitution: GIMB

Mathematics as a Language

Linguistic Functions of MathematicsMathematics serves as a universal language that conveys complex ideas and relationships through symbols and structured syntax. It aids in fostering clear communication and understanding through its precise notation and logical frameworks.

Topics Covered:

• Sets

• Functions

• Binary Operations

• Basic concepts of mathematical language

• Mathematical Logic

• Tools for analysis and communication in mathematics

Mathematical Logic

Important tool for analysis and logical reasoning in mathematicsMathematical logic is essential for validating arguments and proving theorems. It provides tools to express mathematical statements unambiguously, helping to avoid misinterpretations that may arise in verbal or written communication.

Example:Using mathematical logic, we can validate the statement "If it rains, then the ground is wet" through logical reasoning. If we observe it raining and the ground is indeed wet, it confirms that the statement holds true.

Quote by Albert Einstein: "Pure Mathematics is in its way, the poetry of logical ideas"

Introduction to Logic

Definition of Logic:The study of truth and reasoning, focusing primarily on declarative sentences that convey information.

Types of Logic Statements:

• Statement: A declarative sentence that can be evaluated as either true or false (but not both).Examples:

• "Where do you live?" (not a statement)

• "21 + 10 = 45" (false statement)

• "The City of Manila is the capital of the Philippines." (true statement)

• "x ≥ 0" (open statement, truth value unknown without a specific value)

Skills Check on Logic Statements

Instructions:Identify whether given sentences are statements, and if so, determine their truth value:Examples:

• "x + 6 = 5" (open statement, truth value pending)

• "Study your MMW notes." (imperative, not a statement)

• "6 > 0" (true statement)

• "Wow! Math is amazing!" (exclamatory, not a statement)

• "The heart has four chambers: two atria and two ventricles." (true statement)

Symbolic Logic

Representation of Statements:Symbolic logic employs symbols to represent statements and logical relations, enhancing clarity in mathematical communication.

• Simple Statements: Express a single idea, e.g., "The sky is blue."

• Compound Statements: Combine two or more ideas using logical connectives, e.g., "I will stay at home or I will go to the park."

Example:Let p represent "It is raining" and q represent "The ground is wet". The compound statement "It is raining or the ground is wet" can be expressed as p ∨ q.

Truth Values and Truth Tables

Truth Values:Each statement is assigned a truth value, either True (T) or False (F).Truth Tables:Constructing truth tables allows us to determine the truth value of compound statements based on the truth values of their components.

Example:For the statement p → q (If p, then q):

• If p is True and q is True, then p → q is True.

• If p is True and q is False, then p → q is False.

• If p is False, p → q is True regardless of the truth value of q.

Basic Connectives in Logic

• Negation: ~p

• Conjunction: p ∧ q

• Disjunction: p ∨ q

• Conditional: p → q

• Biconditional: p ↔ q

Example:Let p = "I will study" (True) and q = "I will pass the exam" (True).The conjunction p ∧ q would therefore evaluate as True since both components are True.

De Morgan's Laws

Laws of Logic:Demonstrate the relationship between conjunctions and disjunctions:

• ~(p ∧ q) ≡ ~p ∨ ~q

• ~(p ∨ q) ≡ ~p ∧ ~q

Example:If p is "It is raining" (True) and q is "It is cold" (False),then ~(p ∧ q) would be True since both conditions are not simultaneously true, leading to the conclusion that it is either not raining or not cold, or both.

Quantifiers in Logic

• Universal Quantifier (∀): Indicates 'all', e.g., "For every x, P(x) is true."

• Existential Quantifier (∃): Indicates 'some' or 'there exists', e.g., "There exists an x such that P(x) is true."

Negation of Quantified Statements:

• "All X are Y" is negated as "Some X are not Y."

• "No X are Y" is negated as "Some X are Y."

Conclusion

The content focuses on understanding mathematical language through logic, statements, and truth tables, laying the essential groundwork for more complex mathematical concepts.

Practice Exercises

Skills Check Up 04Additional exercises to reinforce understanding of the material!