MMW-CH2

GE MODULE NO. 2: MATHEMATICS IN THE MODERN WORLD

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.

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

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.

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.

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.

Addition (+), Subtraction (-), Multiplication (*), Division (/): Define operations between numbers.
Comparison Symbols:
- = - Equal to
- 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

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 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.

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.

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
  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.

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))

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.

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.

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.