MMW-CH2

GE MODULE NO. 2: MATHEMATICS IN THE MODERN WORLD

MODULE OVERVIEW

• Focus on understanding mathematics as a language with its own symbols, logic, and structure.

• Major areas of understanding include:

  • Basics of mathematics communication (symbols, language, logic).

  • Important features and rules for writing and comprehension of mathematics.

  • Comparison between mathematical expressions and sentences.

  • Standard conventions in mathematical language for consistency and clarity.

  • Introduction to sets, relations, functions, and logic in mathematical arguments.

LEARNING OBJECTIVES

By the end of this module, students should be able to:

• Discuss mathematical language, symbols, and conventions.

• Explain mathematics as a language.

• Correctly perform operations on mathematical expressions.

• Recognize mathematics as a useful language in understanding the world.

INTRODUCTION TO MATHEMATICAL LANGUAGE

• What is Language?

  • Defined as a communication system with sounds, words, and grammar (Cambridge English Dictionary).

  • Mathematics is also considered a language due to its system of symbols and rules.

• Example:

  • Verbal: "Two added to four is six."

  • Mathematical: $2 + 4 = 6$

  • Both forms convey the same idea, illustrating mathematics as a universal language.

CHARACTERISTICS OF MATHEMATICAL LANGUAGE

• Mathematics enables precise, concise, and powerful expression of thoughts.

• Classifications in mathematical language:

  • Nouns (Expressions): Represent mathematical objects (e.g., numbers, variables).

  • Examples:

    • English: person, place, thing

    • Mathematical: $5, 3x - 5, ax + by$

  • Sentences: State complete thoughts (statements, questions, commands).

  • Examples:

    • English: "Pedro is handsome."

    • Mathematical: $2 + 7 = 9$

Nouns vs. Sentences in Mathematical Language

• Expressions (Nouns) do not express complete thoughts.

• Mathematical Sentences (Sentences) can be judged as true or false.

• Verbs in Mathematics: The equal sign $=$ functions as a verb, signifying equality. Connectives such as $+$ link expressions to form compound objects.

LEARNING ACTIVITIES

Learning Activity 1: Identifying Expressions vs. Sentences

• Identify various items as either mathematical expressions or sentences.

Conventions in Mathematical Languages

• Mathematics has conventions helping identify expressions and sentences:

  • Expression: Does not convey complete thoughts.

  • Ex: $3x + 7$

  • Mathematical Sentence: Expresses a complete thought;

  • Ex: $2 + 7 = 9$ states a complete thought and can be evaluated as true.

SYMBOLS AND NOTATION IN MATHEMATICS

Sets and Basic Operations

• Mathematical symbols for sets and operations:

  • $\bigcup$ - Union of sets.

  • $\bigcap$ - Intersection of sets.

  • $<br>i$ - Member of a set.

  • $<br>otin$ - Not a member of a set.

  • $extbf{A} = ext{set label,} ext{ e.g. } A = ext{collection of elements}$

Basic Operations and Relational Symbols

• Addition (+), Subtraction (-), Multiplication (*), Division (/): Define operations between numbers.

• Comparison Symbols:

  • $=$ - Equal to

  • $<br>eq$ - Not equal to

  • $>$ - Greater than

  • $<$ - Less than

  • egin{align} ext{and} \ ext{or} ext{ (logical operations)} ext{ in logical statements.} \ ext{For Ex: } A ext{ and B if A is true and B is true.} \ ext{Equivalents like } A = B ext{ if and only if both are true.} ext{ etc….} ext{ } \ ext{Utilize for defining relationships.}
    ightarrow ext{Express complex relationships clearly through symbols.} ext{ } \ ext{Use Venn Diagrams to illustrate set relations.} \ ext{Illustration of Venn Diagrams showing intersections and unions.} \ ext{Any intersection is common; a union two combined.} ext{ } \ ext{Implications like A implies B, elucidating cause and effect. } \ ext{Logical connectives serve to connect propositions expressively.} ext{Logical connectives guide order of operations in expressions.} ext{ } ext{The combination of expressions yield outputs of logical truth tables.} ext{ } ext{} ext{ } ext{ } ext{} ext{}\ ext{Logic bridges gaps between math, language, and thought processes.}\ ext{Upon benefiting from this bridge, recognizing relations between concepts is essential.} end{align}

• Quantifying Statements:

  • $orall$: For all

  • $hereexists$: There exists

Translation Between English Statements and Symbols

• Translating verbal language into mathematical notation simplifies problem-solving:

  • Example Translating:

  • English: "The sum of six and two over four is two."

  • Mathematical: $\frac{6 + 2}{4} = 2$

Set Properties and Classification

• Set Types:

  • Finite: Countable elements.

  • Infinite: Non-countable elements.

  • Null Set: Set without members, written as $∅$.

  • Unit Set: A set with exactly one element.

• Set Relations:

  • Equivalent: Same number of elements.

  • Equal: Same elements.

Venn Diagrams

• Used to visualize set operations (union, intersection, etc.).

• Demonstrate relationships between sets.

• Example Applications:

  • Assess shared interests among diverse groups.

Cartesian Product

• Defined as all possible ordered pairs from two sets, $A imes B$.

• Example Application: Given sets $A = {1, 2}$ and $B = {3, 4}$, produces pairs: $1, 3; 1, 4; 2, 3; 2, 4$.

• The properties of these sets can be visually represented through set diagrams, refining perspectives of set relationships.

B. THE LANGUAGE OF RELATIONS AND FUNCTIONS

• Involvement of ordered pairs shows relationships between two sets.

  • A function defines a relation without ambiguity or duplicity in outputs: every input relates to a single output.

  • Example: Function $f: X <br>ightarrow Y$ means for each $x ext {in } X$, the function provides one $y ext {in } Y$.

• Relation Properties:

  • Reflexive: All elements relate to themselves.

  • Symmetric: Relation remains consistent if reversed.

  • Transitive: Establishes continuity through relationships.

  • Antisymmetric: Equality holds with respect to the property.

Function Operations

• Basic operations on functions include:

  • Sum, Difference, Product, Quotient

  • Composition of Functions

• Example with functions for clarity:

  • $f(x) + g(x)$ or $f(g(x))$

C. LOGIC AND FORMALITY IN MATHEMATICS

• Logic serves as the backbone for reasoning in mathematics.

  • Propositions can either be true or false.

  • Examples include:

  • True: "2 is even."

  • False: "The integer 3 is even."

• Truth Tables:

  • Construct tables to explore logical connectors: conjunctions, disjunctions, implications, and equivalences.

ASSESSMENT OPPORTUNITIES

• Engage in practical activities translating verbal statements into mathematical symbols and vice versa to solidify understanding.

• Activities also include writing and evaluating propositions, completing truth tables for varying conditions and operations in mathematical contexts.

REVISION AND REFLECTION

• Regularly reassess learned concepts in math and logic, deepen connections between various domains of study and application.

• Engage insights derived from formal mathematical concepts in practical scenarios.