Language of Mathematics and Mathematical Logic

Objectives and Overview of Mathematical Language

• Key Objectives of Mathematical Language Study:

  • Discuss the language, symbols, and conventions of mathematics.

  • Explain the nature of mathematics as a language and acknowledge its utility in real-world application.

  • Compare and contrast mathematical expressions and mathematical sentences.

  • Identify and discuss the four basic concepts in mathematical language.

  • List and discuss basic operations on logic and logical formalities.

  • Perform operations on mathematical expressions correctly.

  • Articulate the importance of mathematics in one's life and express appreciation for mathematics as a human endeavor.

• Group Presentation Details:

  • Presented by: Group 1 MMW (BSN 1-E Group 1 MMW)

  • Group Members: Alabat, Sienna Bernardine; Arejola, Guelda; Cala, Rogel Angelo; Chavez, Cris; De Vivas, Junessa Lyn; Lardizabal, Deinah Shannel; Manansala, Elyza; Oafallas, Rona

• Core Equations Expressed in Mathematical Notation:

  • Point-Slope Form Equation: $y - y_1 = m(x - x_1)$

  • Standard Quadratic Equation: $ax^2 + bx + c = 0$

  • Acceleration Formula: $a = \frac{V_f - V_i}{t}$

Characteristics and Conventions of Mathematical Language

• Definition of Mathematical Language:

  • A systematic way of expressing mathematical ideas using numbers, symbols, words, and rules.

• Four Primary Characteristics of Mathematical Language:

  • Precise: Able to make very fine distinctions and convey exact, specific meanings.

  • Concise: Able to express long or complex ideas in a short, brief manner.

  • Efficient: Able to communicate ideas clearly and directly with minimal friction.

  • Powerful: Able to express complex and abstract mathematical ideas effectively.

• Common Mathematical Conventions:

  • Mathematical formulas are traditionally written from left to right.

  • The Latin alphabet is standardly utilized for variables and parameters, such as $x$, $y$, $a$, and $b$.

Mathematical Vocabulary, Special Terms, and Taxonomy

• Mathematical Terms with Specific Definitions:

  • Group: A set of mathematical objects combined with a specific operation.

  • Ring: A mathematical algebraic system equipped with addition and multiplication operations.

  • Field: A algebraic system in which addition, subtraction, multiplication, and division can all be performed.

  • Term: A number, variable, or combination of numbers and variables.

  • Factor: Numbers or mathematical expressions that are multiplied together to yield a product.

• Special Advanced Mathematical Vocabulary:

  • Tensor: A mathematical object used to describe linear relationships between physical or algebraic quantities.

  • Fractal: A repeating mathematical pattern that displays self-similarity across different scale sizes.

  • Functor: A structural map in category theory used to connect different mathematical structures.

• Mathematical Taxonomy (Hierarchy of Logical Statements):

  • Axiom: A fundamental statement accepted as true without proof.

  • Conjecture: An idea or proposition believed to be true based on observation but not yet mathematically proven.

  • Theorem: A mathematical statement that has been formally proven to be true using logic and established axioms.

  • Lemma: A proven auxiliary statement used as a stepping stone to assist in proving a larger theorem.

  • Corollary: A direct structural result or proposition that follows immediately from a proven theorem.

Mathematical Symbols and Notations

• Basic Arithmetic and Equivalence Symbols:

  • Plus: $+$

  • Minus: $-$

  • Times (Multiplication): $\times$

  • Division: $\div$

  • Equal to: $=$

  • Not Equal to: $\neq$

  • Nearly Equal to: $\approx$

  • Is Congruent to: $\cong$

  • Identical to: $\equiv$

  • Equivalent: $\equiv$

• Relational and Ordering Symbols:

  • Less than: <

  • Greater than: >

  • Less than or equal to: $\le$

  • Greater than or equal to: $\ge$

  • Much less than: $\ll$

  • Much greater than: $\gg$

• Geometric, Relational, and Quantifier Symbols:

  • Parallel to: $\parallel$

  • Perpendicular: $\perp$

  • Proportional to: $\propto$

  • Angle: $\angle$

  • Therefore: $\therefore$

  • Implies: $\implies$

  • For all (Universal Quantifier): $\forall$

  • There Exists (Existential Quantifier): $\exists$

  • There does not exist: $\nexists$

  • Factorial: $!$

  • Ampersand: $\&amp;$

  • Ellipsis: $\dots$

• Set Theory, Calculus, and Operational Symbols:

  • Element / Member of: $\in$

  • Contains as Member: $\ni$

  • Union: $\cup$

  • Intersection: $\cap$

  • Summation: $\sum$

  • Integral: $\int$

  • Partial Derivative: $\partial$

  • Nabla / Vector Differential Operator: $\nabla$

  • Delta / Increment: $\Delta$

  • Infinity: $\infty$

  • Absolute Value: $|x|$

  • Percentage: $\%$

  • Tensor Product: $\otimes$

• Radicals, Units, and Greek Letters:

  • Square Root: $\sqrt{x}$

  • Cube Root: $\sqrt[3]{x}$

  • Fourth Root: $\sqrt[4]{x}$

  • Degrees: $^\circ$

  • Degrees Celsius: $^\circ\text{C}$

  • Degrees Fahrenheit: $^\circ\text{F}$

  • Alpha: $\alpha$

  • Beta: $\beta$

  • Gamma: $\gamma$

  • Epsilon: $\epsilon$

  • Theta: $\theta$

  • Mu: $\mu$

  • Pi: $\pi$

  • Rho: $\rho$

  • Sigma: $\sigma$

  • Tau: $\tau$

  • Phi: $\phi$

  • Omega: $\omega$

Mathematical Expressions versus Sentences

• Mathematical Expression:

  • Definition: A correct, valid arrangement of mathematical symbols.

  • Components: Contains numbers, variables, operations, and functions.

  • Complete Thought: Does NOT express a complete thought.

  • Truth Value: Cannot be assigned a truth value; it cannot be judged as true or false.

• Mathematical Sentence:

  • Definition: A mathematical statement that compares, relates, or equates two mathematical expressions.

  • Relational Symbols Used: Employs relational symbols such as $=$, >, and <

  • Complete Thought: Expresses a complete mathematical thought.

  • Truth Value: Can be evaluated as TRUE, FALSE, SOMETIMES TRUE, or SOMETIMES FALSE.

Four Basic Concepts in Mathematical Language

• The Four Fundamental Foundations:

  • Language of Sets

  • Language of Functions

  • Language of Relations

  • Language of Binary Operations

Language of Sets

• Basic Definitions and Elements:

  • Set: A well-defined collection of distinct objects.

  • Elements / Members: The individual objects forming a set.

  • Belongs to Symbol ($\in$): Indicates that an object is an element of a set.

  • Does not belong to Symbol ($\notin$): Indicates that an object is not an element of a set.

  • Concrete Example: Given set $A = \{1, 2, 3, 4, 5\}$, the element $2 \in A$ (2 belongs to $A$), whereas $7 \notin A$ (7 does not belong to $A$).

• Historical Founder of Set Theory:

  • Georg Ferdinand Ludwig Philipp Cantor (Georg Cantor): German mathematician recognized as the founder of set theory as a distinct mathematical discipline.

  • Birth Date and Place: March 3, 1845 in Saint Petersburg.

  • Death Date and Place: January 6, 1918 in Halle.

• Methods of Expressing Sets:

  • Roster Method (Tabulation Method): Lists each individual element of the set separated by commas within braces. Example: $E = \{a, e, i, o, u\}$.

  • Rule Method (Set-Builder Notation): Defines the set by describing the rule or common property satisfied by all elements, written in the general form $\{x \mid P(x)\}$ where the bar $\mid$ denotes "such that". Example: $E = \{x \mid x \text{ is a vowel letter}\}$ (E is the set of all $x$ such that $x$ is a vowel letter).

• Classifications and Terms on Sets:

  • Finite Set: A set with a countable, limited number of elements whose elements can be identified completely. Examples: $A = \{x \mid x \text{ is a positive integer less than } 10\} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$; $E = \{a, e, i, o, u\}$.

  • Infinite Set: A set containing elements that extend indefinitely without end, making it impossible to identify a final element. Examples: $F = \{\dots, -2, -1, 0, 1, 2, \dots\}$; H = \{x \mid x \text{ is a set of molecules on earth\}.

  • Unit Set (Singleton): A set containing exactly one single element. Examples: $I = \{x \mid x \text{ is a whole number greater than } 1 \text{ but less than } 3\} = \{2\}$; $K = \{\text{rat}\}$.

  • Empty Set (Null Set): A set containing zero elements, represented symbolically by $\emptyset$ or $\{\}$. Examples: $L = \{x \mid x \text{ is an integer less than } 2 \text{ but greater than } 1\}$; N = \{x \mid x \text{ is the set of positive integers less than zero\}.

  • Universal Set ($U$): A fixed set containing all objects or elements under consideration in a given discussion or set theory application. Example: $U = \{x \mid x \text{ is a positive integer}, x^2 = 4\}$.

  • Cardinality: The exact count of elements contained within a set, denoted as $n(A)$. Example: For set $E = \{a, e, i, o, u\}$, the cardinality is $n(E) = 5$.

• Venn Diagrams:

  • Definition: A visual graphical tool used to represent relationships and logical connections among finite groups of sets using simple closed curves, typically overlapping circles.

  • Structure: The region inside a circle represents elements inside the set, while the region outside represents elements not included in the set.

  • Origin: Introduced by John Venn in his landmark paper "On the Diagrammatic and Mathematical Representation of Propositions and Reasonings."

• Types and Relations Between Sets:

  • Subset: Set $A$ is a subset of $B$ if and only if every element contained in $A$ is also an element of $B$. Symbolically: $A \subseteq B \iff \forall x, (x \in A \rightarrow x \in B)$.

  • Proper Subset: Set $A$ is a proper subset of $B$ if all elements of $A$ are present in $B$, and set $B$ contains at least one additional element not found in $A$. Symbolically: $A \subset B$.

  • Equal Sets: Set $A$ equals set $B$ if and only if every element of $A$ is in $B$ and every element of $B$ is in $A$. Symbolically: $A = B \iff (A \subseteq B \land B \subseteq A)$.

  • Power Set: The comprehensive set containing every possible subset of a given set $S$, denoted as $\mathcal{P}(S)$.

• Operations on Sets:

  • Union: The union of sets $A$ and $B$, denoted $A \cup B$, is the set of all elements $x$ in universal set $U$ such that $x$ belongs to $A$ or $x$ belongs to $B$. Symbolically: $A \cup B = \{x \mid x \in A \lor x \in B\}$.

  • Intersection: The set containing all common elements that belong simultaneously to both set $A$ and set $B$, denoted $A \cap B$.

  • Complement (Absolute Complement): The complement of $A$, denoted $A'$, is the set of all elements in universal set $U$ that are not elements of $A$. Symbolically: $A' = \{x \in U \mid x \notin A\}$.

  • Difference (Relative Complement): The difference of $A$ and $B$ (relative complement of $B$ with respect to $A$), denoted $A \sim B$ or $A \setminus B$, is the set of elements $x$ that belong to $A$ but do not belong to $B$. Symbolically: $A \sim B = \{x \mid x \in A \land x \notin B\} = A \cap B'$.

  • Symmetric Difference: The set of elements that belong to either set $A$ or set $B$, but not to both.

  • Disjoint Sets: Sets that have no elements in common, meaning their intersection is the empty set ($A \cap B = \emptyset$).

  • Ordered Pairs: Paired mathematical entities $(x, y)$ where the precise sequence of components carries operational meaning.

Language of Relations and Functions

• Concept of Relations:

  • Relation: A collection or set of ordered pairs $(x, y)$. When a relation exists between elements $x$ and $y$, $x$ corresponds to $y$ and $y$ depends on $x$.

  • Relation from Set $A$ to Set $B$: Defined as any subset of the Cartesian product $A \times B$. If $(a, b) \in R$, $a$ is related to $b$, written $a R b$

  • Domain of $R$ ($\text{dom } R$): The set of all first components $a \in A$ from the ordered pairs in relation $R$. Symbolically: $\text{dom } R = \{a \in A \mid (a, b) \in R \text{ for some } b \in B\}$.

  • Image / Range of $R$ ($\text{im } R$): The set of all second components $b \in B$ from the ordered pairs in relation $R$. Symbolically: $\text{im } R = \{b \in B \mid (a, b) \in R \text{ for some } a \in A\}$.

• Concrete Examples of Relations:

  • Car Brand and Manufacturer Country Pairing: Let $A = \{a, b, c, d\}$ represent car brands and $B = \{s, t, u, v\}$ represent manufacturing countries. The Cartesian product $A \times B$ yields all possible pairings. Relation $R$ from $A$ to $B$ is defined as $R = \{(a,s), (a,t), (a,u), (a,v), (b,s), (b,t), (b,u), (b,v), (c,s), (c,t), (c,u), (c,v), (d,s), (d,t), (d,u), (d,v)\}$.

  • Even Sum Parity Relation: Let $A = \{4, 7\}$. The Cartesian product A \times A = \{(4,4), (4,7), (7,4), (7,7)\}$. Defining relation R$on$A$as$x \sim y \iff x + y \text{ is even}$, the valid ordered pairs are$(4,4) \in R$(since$4+4=8$) and$(7,7) \in R$(since$7+7=14).\n\n* Language of Functions:\n * Function Definition: A special type of mathematical relation where every single input element in the domain corresponds to exactly one unique output element in the range.\n * Domain (X): The entire set of input values.\n * Range / Codomain (Y$): The set of output values produced by applying rule$f(x).\n * Function Mapping: A rule or mapping f(x) connecting input values to outputs, where the range represents the set of all images of domain elements.\n\n# Language of Binary Operations and Group Theory\n\n* Algebraic Foundations:\n * Algebraic Structure: A mathematical set of elements combined with an operation that satisfies defined mathematical rules or axioms (\text{Set} + \text{Operation} + \text{Rules} = \text{Algebraic Structure}).\n * Binary Operation: An operation that combines two elements from a set to generate a single resulting element that belongs to the same set.\n * Group: A fundamental algebraic structure consisting of a set G equipped with one binary operation that satisfies four mandatory properties.\n\n* Four Fundamental Properties of a Group:\n * Closure Property: Combining any two elements in the set via the operation results in an element that is also in the set. Symbolically: a * b = c \in G$for all$a, b, c \in G\n * Associative Property: Rearranging the parentheses in an operational sequence does not alter the final result. Equations: (a + b) + c = a + (b + c)$and$(a \times b) \times c = a \times (b \times c)$, or$a * (b * c) = (a * b) * c\n * Identity Property: There exists a distinct identity element e$in the set such that operating any element$a$with$e$leaves the value of$a unchanged.\n * Inverse Property: Every element a$in the set possesses a unique inverse element$a^{-1}$such that operating$a$with$a^{-1}$produces the identity element$e, thereby allowing operations to be inverted or undone.\n\n* Set of Group G Properties:\n * A valid group G$contains all operational results and adheres strictly to closure, associativity, identity$e$, and inverse$a^{-1}.\n * Operational Group Test Example: Determining whether non-negative integers under addition form a group requires applying closure. Testing 8 + 4 = 12$and$5 + 10 = 15 confirms closure because adding two non-negative integers always yields a non-negative integer.\n\n# Formal Logic and Mathematical Logic\n\n* Formal Logic:\n * Definition: The formal science or study of evaluating arguments, deductive structures, and logical reasoning.\n * Primary Function: Distinguishes correct and sound reasoning from flawed or poor reasoning, providing structured methods for clear thought.\n\n* Mathematical Logic:\n * Definition: A specialized branch of mathematics that studies theoretical logic, formal reasoning systems, deductive formal proof systems, and expressive formal systems.\n\n* Four Major Divisions of Mathematical Logic:\n * Set Theory: Studies sets as collections of objects, analyzing how sets are formed, how they interact, and how they combine. Example: Sets A = {1, 2, 3}$and$B = {3, 4, 5}.\n * Recursion Theory: Studies mathematical problems, processes, and computable functions that can be solved step-by-step using algorithmic rules. Example: Sequential process 1, 2, 3, 4, 5, \dots\n * Proof Theory: Studies formal mathematical proofs, examining how logical inference rules demonstrate the truth of statements. Example: All even numbers are divisible by 2$;$8$is an even number; therefore,$8$is divisible by$2$$

Model Theory: Studies formal mathematical structures (models) to evaluate whether specific logical statements or operational rules hold true within those models. Example: Given a system defined by rules, model theory evaluates: "Does this model satisfy these rules?"