Mathematical Language and Symbols

• Discusses mathematical language and symbols
• Specifically, the characteristics of mathematical language, its similarities and differences from ordinary languages, and its basic syntax and semantics
• Understanding formal definitions of sets, relations, functions, and binary operations is an objective
• Contains three topics: The Language of Mathematics, Elementary Logic, and Basic Concepts of Mathematics
• There is an Activity section at the beginning of each topic
• Illustration sections help you understand concepts
• Practice exercises can be found in the "Try this!" sections on pages 52 and 66-67
• Answers are provided in the "Check your work!" section on pages 68-69
• A Summative Test is provided at the end of the module
• The objectives are to accurately do each of the following:
• Discuss the three characteristics of math language
• Cite similarities and differences between math language and other ordinary language
• Determine the truth value of a given compound proposition and a proposition involving quantifier
• Express quantifications in both math symbols and in English language
• Express mathematical sentences involving sets, relations, functions, and binary operations in mathematical symbols and in English
• Solve problems involving sets, relations, function and binary operations

Lesson 1: The Language of Mathematics

• Mathematics can be seen as a language
• Learning mathematics effectively requires understanding its language by knowing its vocabulary and grammar rules.
• One’s ability to communicate through this language is greatly enhanced by knowing the basic grammar rules of the language
• Understanding the symbols how they relate to each other is also important
• Similar to ordinary language, mathematical language has similarities to English:
• Alphabet/Word: Mathematical Symbol/Concept (e.g., +, =)
• Noun: Expression (e.g., 10, x)
• Sentence: Mathematical Sentence (e.g., 10 = x + 13)
• "Mathematical concept or mathematical symbol” is a counterpart of word and each symbol/concept has its corresponding meaning.
• This meaning is expressed via definitions, postulates, and theorems.
• "Mathematical expression" is the counterpart of a "noun" and mathematical expressions are names given to a mathematical object of interest
• An expression like '10 + x' can also be written as 'x + 10' due to the commutative rule of addition, but it cannot be written as 10x + or +10x.
• "Mathematical sentence" is the counterpart to "sentence"
• Mathematical sentences refer only to declarative sentences that can be true or false (called propositions)
• A mathematical sentence is a correct arrangement of mathematical symbols that states a complete thought (e.g., 10 + x = 13)
• Specific rules must be followed when forming a mathematical sentence based on the mathematical concepts used
• Characteristics that differentiate math from ordinary languages are the following:
• Precise: Able to make very fine distinctions
• Concise: Able to express things briefly
• Powerful: Able to express complex thoughts with relative ease
• Precision in math prevents errors that can occur from a lack of vagueness and ambiguities unlike in ordinary languages
• Math uses both English and mathematical symbols to express ideas
• For example, "x is an element of the set of positive integers" can be written as x ∈ Z+, a mathematical expression to mean that x is a positive integer
• x ∈ Z⁺ can be understood by different nationalities who know the language of math
• Math provides formulas for complex problems that create solutions

Lesson 2: Elementary Logic

• Syntax refers to the rules to be followed for describing "grammatically correct" sentences
• Semantics refers to the study of meaning
• Studying elementary logic helps understand the rules of grammar and meaning in mathematical language
• A proposition (or statement) is a declarative sentence that is either true or false, but not both
• Understanding propositions is important in the study of mathematics because mathematical concepts are described in the form of definitions, postulates, theorems, etc.
• Math is concerned with determining whether propositions are true or false, which is their truth value
• The truth value of a proposition is true (T) or false (F)
• Propositions are expressed using small letters (p, q, r, s, t) called propositional variables Sentences like "What a wonderful day!" are not propositions; however, sentences like "Tacloban is a highly urbanized city" are becauase they are declarative
• A sentence like 'x-1=6' is declarative, but it is not a proposition because a variable is assigned
• Propositional variables can be used to denote sentences
• Compound propositions are formed from given propositions using logical connectives including not, and, or, if ... then, and if and only if
• The five types of compound propositions are negation, conjunction, disjunction, conditional statement, and biconditional statement
• Simple statements being combined should relate to each other
• Table of the five types of compound propositions:
• Negation: "not" connective; "~p" symbolic form; "It is not the case that p" English statement; the truth value of ~p is the opposite of the truth value of p
• Conjunction: "and" connective; "p ∧ q" symbolic form; "p and q" English statement; The truth value of p ∧ q is true when both p and q are true, otherwise, its truth value is false
• Disjunction: "or" connective; "p V q" symbolic form; "p or q" English statement; The truth value of p V q is false when both p and q are false, otherwise, its truth value is true
• Conditional Statement: "if ... then" connective; "p → q" symbolic form; "If p, then q" English statement; The truth value of p → q is false when the premise p is true and the conclusion q is false, otherwise, its truth value is true
• Biconditional Statement: "if and only if" connective; "p ↔ q" symbolic form; "p if and only if q" English statement; The truth value of p ↔ q is true when p and q have the same truth values, otherwise, its truth value is false

Quantifiers

• Declarative sentences with variables can be true or false depending on the variable, which is why they are not categorized as a proposition, but quantification is used to resolve this
• There are two ways of doing quantification: assigning a specific value to the variable and using quantifiers such as the universal and the existential quantifier

Illustration 9. Quantification by assigning specific value of the variable

• Defining the variable can change the declarative sentence into a true proposition
• For example, if x = 7, then the declarative sentence x - 1 = 6 becomes 7 - 1 = 6, which is a true proposition
• But If the defined variable creates a false proposition , it is called a counterexample

Illustration 10. Quantification by using the universal quantifier

• A variable is defined for a particular set of values and called "the domain of definition or simply called the domain
• "∀ x ∈ A, x − 1 = 6" is "the mathematical sentence".
• The symbol ∀ is the universal quantifier read as "for all" or "for every"
• The truth value of a universal quantification is true if all elements are examples otherwise there is a counterexample

Illustration 12. Quantification by using the existential quantifier

• An existential quantification mathematical symbol is "∃x ∈ A, x − 1 = 6."
• Translation: "For some element x in set A, x minus one is equal to six", or "There exists an element x in set A such that x minus one is equal to six".
• The symbol ∃ is called an existential quantifier
• The truth value of an existential quantification is true if there is an example found in the domain of definition but if there is no example its false.

Lesson 3: Basic Concepts of Mathematics: Sets, Relations, Functions and Binary Operations

• Basic concepts of mathematics (sets, functions, relations, and binary operations) are formally defined using mathematical language
• The study of basic concepts is inevitable because they a foundation of math concepts

Sets

• "a set is a collection of well-defined objects", which can be denoted by roster method or set-builder notation and elements are denoted by capitol letters
• Each object found in the set is called an element of the set
• If an object x belongs to set A, it is written as x ∈ A (x is an element of set A)
• If an object does not belong to set A, it is written as x ∉ A (x is not an element of set A)
• The union of two sets A and B ( AUB ) is defined as AUB = {x|x ∈ A V x ∈ B)
• A union forms a set, therefore all objects x that will not make the declarative sentences "x is an element of A" and "x is an element B" both false is collected
  • That is object x may belong to set A alone, or to set B alone, or it may belong to both
• Intersection of two sets AB = is set of all x A and B that makes the declarative sentence "x is an element of A and x is an element B" or the A set and and B subset.

Relations

• Relation in Math are between #s to to binary operations
• Binary relation R from A to B is a subset of A × B where it only involves two sets or the relation describing how set A is related to set B.
• Intuitively, Relation R should contain some or all of these ordered pairs (x, y) in A × B

Functions

• Types of relations pair each first set to a unique element of the 2nd
• A functions from est A to Set B means denoted by F:A -> B, A to B where each x = A There exists Unique Y where y = f(x). (ordered pairs).
• Then it becomes a true propostion when x is substituted by x. If the sentence is true it will provide the Y subset B for X.
• Must find Element of B that substitutes all requirements.

Binary operations

• Addition, subtraction, multiplication, division of numbers, composition of functions, vector addition, and a lot more
• Binary operation on A is function from A × A to A were A has been already set
• The set ordered pairs (x,y,y) such that x an y are elements of A
• Binary operation assigns every ordered pair of of a, composed two elements of set, each element should be set.