MATHEMATICAL LANGUAGE AND SYMBOLS

MODULE 2 - MATHEMATICAL LANGUAGE AND SYMBOLS

MATHEMATICAL LANGUAGE
Mathematical language is a system used to communicate mathematical ideas, concepts and theories. It has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, idioms, abbreviations, sentence structure and paragraph structure. Mathematical language is characterized by: abstraction, symbols and rules, non-linearity and complexity of language, arrangement, coding, and decoding information. While language was invented to communicate ideas to others, mathematical language was designed for numbers, sets, functions, and perform operations.

MATHEMATICAL SYMBOLS

fhfgh

CHARACTERISTICS OF MATHEMATICS
According to Jamison (2000) the use of language in mathematics differ from language of ordinary speech in three important ways:

Non-temporal - there is no future, present and past tense
Devoid of emotional content
Precise

According to Dr. Carol Burns, mathematics language is:

Precise – able to make very fine distinction
Concise – able to say or write things briefly
Powerful – able to express complex thoughts with relative ease

EXPRESSION VS SENTENCE
A mathematical expression is a mathematical analogue of an English noun. It does not state a complete thought. The most common expressions are numbers, variables, sets and functions. It cannot be determine if it is true or false. Examples of mathematical expression are: x + 4, 8x, x/2, etc.
Meanwhile, a mathematical sentence is a mathematical analogue of an English Sentence. It states a complete thought. It is always used relation symbols such as =, ≠, >, <, ≥, or ≤. Usually, it has verbs. Examples of mathematical sentence are: x + 4 = 7, 8x > 7, etc.

TYPES OF MATHEMATICAL SENTENCE
1. Open sentence - uses variables, it is not know whether or not the mathematical sentence is true or false. (Sometimes True/Sometimes false) Ex.: 2xy < 3y
2. Close sentence - is known to be either true or false. (True Closed Sentence - TCS and False Closed Sentence - FCS) Ex.: 2 + 1 = 3 (TCS); 2 + 1 = 1 (FCS)

EQUATION VS INEQUALITY
An equation is a mathematical statement that shows the equal value of two expressions while an inequality is a mathematical statement that shows that an expression is lesser than or more than the other. An equation shows the equality of two variables while an inequality shows the inequality of two variables. Although both can have several different solutions, an equation only has one answer while an inequality can also have several.

GRAMMAR OF MATHEMATICS
Grammar of Mathematics is the mathematical notation used for formulas has its own grammar, not dependent on a specific natural language, but shared internationally by mathematicians regardless of their mother tongues. Mathematics can be consider as a common language of the world because it is adopting a universally understood symbolic system for its language.
1. The word “is” could mean equality, inequality, or membership in a set depending on how they are used in a sentence.2. A number in a sentence may be of cardinal, ordinal, or nominal type.3. The words “and” and “or” mean differently in mathematics from its English use.4. Mathematical objects may be represented in many ways such as sets and functions.

MATHEMATICAL CONVENTION
Mathematical convention is an accepted rules and practice of spelling, writing and punctuations. It is a fact, name, notation or usage which is generally agreed upon by mathematicians. It is also a technique used by mathematicians, engineers, scientists, in which each particular symbols has a particular meaning.
1. Numbers - are symbols used in counting, constant values which can be both real numbers and imaginary numbers.2. English Alphabet - usually represents the possible values for quantities. They are commonly used as variable. In the study of sets, capital letters serve as names of the sets and the lowercase letters stand for the elements of those sets.3. Greek Alphabet - usually represents constants formulas in mathematics, physics, chemistry, and other related branches.4. Symbols - express operations or relationships.

FOUR BASIC CONCEPTS OF MATHEMATICAL LANGUAGE

1. LANGUAGE OF SET
Set is a prescribed collection of points, numbers, or other objects that satisfy a given condition. It is a collection of objects or things. The objects or things in the set are called elements (or members) of the set. It is denoted by a capital letter. George Cantor introduced the study of sets in 1870's. Further, set theory is the branch of mathematics that studies sets or the mathematical sentence of the infinite. The following two methods are often used to designate a set. The third and last method is used to generalize a given set using symbols.

Describe the set using words
Roster method – also called Listing notation, is when the elements of the set are enumerated and separated by a comma {,}. For example: {1, 2, 3, 4}
Rule method - also called set builder notation, symbol written as {x│P(x)}. It involves specifying a rule or condition which can be use to decide whether anobject can belong to the set.

BASIC PROPERTIES OF SET

Empty Set or Null Set - is the set that contains no elements. The symbol Ø or { } is used to represent the empty set. Example: the set of natural numbers that are negative integers
Finite Set - if the number of elements in the set is a whole number. A set whose elements are limited or countable. Example: A = {1, 2, 3}
Cardinal Number of finite set - is the number of elements in the set. It is denoted by the notation n(A) where A is a set. Example: S = {1, 2, 3, 4} then n(S) = 4
Equal Set - Set A is equal to Set B, denoted by A = B, if and only if A and B have exactly the same elements. Example: {a, b, c} = {b, c, a}
Equivalent Set - Set A is equivalent to Set B, denoted by A ~ B, if and only if A and B have the same number of elements. Example: {a, b, c} ~ {1, 2, 3}
Universal Set - denoted by U, is the set of all possible elements of any set used in the problem. The set of all elements that are being considered.

BASIC SET OPERATORS

The Complement of a Set - the complement of a set A, denoted by A', is the set of all elements of the universal set U that are not elements of A. In order to determine the complement of set A, you must know the elements of A as well as the elements of the universal set U that is being used. Note: The complement of the universal set is the empty set and the complement of the empty set is the universal set. (U' = Ø and Ø' = U)
Cardinal Number Formula for the complement of a set - For any set A and its complement A’: n(A) + n(A') = n(U)
Subsets - Set A is a subset of Set B, denoted by A ⊆ B, if and only if every element of A is also an element of B. Two fundamental subset relationships:

A ⊆ A for any set A (the set itself is a subset of that set)
Ø ⊆ A for any set A (an empty set is a subset of any set)

Proper Subsets - Set A is a proper subset of Set B, denoted by A ⊂ B, if every element of A is an element of B, and A ≠ B.
The Number of Subsets of a Set - A set with n elements has 2ⁿ subsets. Example: {1, 2, 3, 4, 5, 6} has 6 elements, so it has 2⁶ = 64 subsets.
Superset - Set A is a superset of Set B, denoted by A ⊇ B, if A has same element as B or more. Example: {1, 2, 3} ⊇ {1, 2, 3}
Proper Superset - Set A is a proper superset of Set B, denoted by A ⊃ B, if A has B's elements and more. Example: {1, 2, 3, 4} ⊃ {1, 2, 3}
Intersection of Sets - The intersection of sets A and B, denoted by A ∩ B, is the set of all elements common to both A and B.
Union of Sets - The union of sets A and B, denoted by A ∪ B, is the set that contains all the elements that belongs to A or to B, or to both.

Take note: ~ The word "and" is interpreted to mean intersection. The phrase "the elements of A and B" means A ∩ B.~ The word "or" is interpreted to mean union. The phrase "the elements of A or B" means A ∪ B.

Cardinal Number Formula for the Union/Intersection of Sets - For any two sets A and B, the number of elements in their union is n(A ∪ B) where: n(A∪ B) = n(A) + n(B) - n(A ∩ B) and n(A ∩ B) is the number of elements in their intersection. "As long as any three of the four quantities in the general formula are known, the missing quantity can be found by algebraic manipulation.
Singleton Set - is the set that contains one element. Cardinality is equal to one. Example: A = {1}
Infinite Set - a set with unlimited elements. Cardinality is infinite. Example: Set N is the set of counting numbers, N = {1, 2, 3, 4, 5, ...}
Power Set - the set of all the subsets of a set. Example: A = {1, 2} then P(S) = { }, {1}, {2}, {1,2}. Thus cardinal number N(P(S)) = 2² = 4 subsets

2. THE LANGUAGE OF RELATION
Relation pertains to the relationship between two or more variables. It is a collection of pair of information in such a way that for every input there is a single or multiple corresponding output. In Mathematics, relations are sets of ordered pairs, (x, y) where x is the domain, input, and independent variable while y is the range, output, and dependent variable. Further, the concept and theory of relations was introduced by Augustus De Morgan.

TYPES OF RELATION
1. One to One Relation - Every input has a corresponding unique output. Example: (-3,-2), (-1,0), (2,4)
2. One to Many Relation - Every input has multiple outputs. Example: (-1,0), (-1,1), (-1,2)
3. Many to One Relation - Multiple inputs with the same output. Example: (1, 0), (2,0), (3,0)

3. LANGUAGE OF FUNCTION
Function is a correspondence, or relationship, between two sets called the domain and range such that for each element of the domain there corresponds exactly one element of the range. It is a set of ordered pairs (x, y) such that no two ordered pairs have the same x – value but different y – values. Function notation is frequently used for equations that define function. The letter f is commonly used to name a function such as f(x).

WAYS OF REPRESENTING A FUNCTION

By equation in two variables - In this method, you are given an equation with two variables, one is independent and one is dependent. What you will do is assign values for the independent variable, then substitute it into the equation to predict the dependent variable. For example, in the equation y = 2x, when x = 2, y = 4.
By mapping - A mapping shows how the elements are paired. Its like a flow chart for a function, showing the input and output values. A mapping diagram consists of two parallel columns. The first column represents the domain of a function f , and the other column for its range.
By table - In a table, you will have columns and rows. Usually, a two-column table is used in which the first column will be the independent variable while the second column will be the dependent variable.

By graph - Graph uses a coordinate plane in which the functions will be plotted. Note that the x-axis will be the independent variable and the y-axis will be the dependent variable.
By ordered pairs - The functions will be in the form (x, y) where x is the independent variable and y is the dependent variable.

"All functions are relation, but not all relations are function."3 Types of Relations1. One to one Relation - Function2. One to many Relation - Not Function3. Many to one Relation - Function
How to determine whether a relation is a function or not?

Focus on the x – coordinate or domain, when given a relation.

✔ If the set of ordered pairs has different x – coordinate or domain, it is a function✔ If the set of ordered pairs has same x – coordinate or domain, it is NOT a function✔ y – coordinate or range have no bearing in determining functions

Vertical Line Test - A relation is a function if a vertical line drawn through its graph, passes through only one point.

EVALUATING A FUNCTION
Evaluating a function means finding the value of f(x) =… or y =… that corresponds to a given value of x. To do this, simply replace all the x variables with whatever x has been assigned. For example, if we are asked to evaluate f(4), then x has been assigned the value of 4. Find the f(4) of f(x) = 2x - 1. Substituting 4, we got 2(4) - 1 = 8 -1 = 7. Thus, the ordered pair is (4, 7).
TYPES OF FUNCTION BASED ON SET ELEMENTS

One to One Function - Bijective - A function is bijective if for every y in the codomain B is exactly one x in the domain.
Many to One Function - Constant
Onto Function - Surjective - A function is surjective if for every y in the codomain B is at least one x in the domain.
Into Function - Injective - A function is injective if for every y in the codomain B is at most one x in the domain. Not all elements in the codomain must be included in the range.

4. BINARY OPERATIONS
A binary operation, binary means two quantities, is any operation that combines two values to create a new one. Common binary operations are addition, subtraction, multiplication, and division. A binary operation can be considered as a function whose input is two elements of the same set S and whose output also is an element of S. Two elements a and b of S can be written as a pair (a, b) of elements in S. As (a, b) is an element of the Cartesian product S × S we specify a binary operation as a function from S × S to S.We use symbols to represent binary operations instead of function names, just as we do with addition and multiplication of integers. Addition uses the symbol + and multiplication uses the symbol ×. We will use symbols such as ⋆ and ∙ to represent arbitrary (non-specific) binary operations, and we will also define new binary operations using the symbols ⊕ and ⊗.

PROPERTY OF BINARY OPERATIONS

Closure Property

A binary operation * on a non-empty set P has closure property, if a ∈ P, b ∈ P ⇒ a * b ∈ P. For example, addition is a binary operation that is closed on natural numbers, integers, and rational numbers.

PROPERTY OF BINARY OPERATIONS

Associative Property

The associative property of binary operations holds if, for a non-empty set S, we can write (a * b) *c = a*(b * c), where {a, b, c} ∈ S. Suppose Z be the set of integers and multiplication be the binary operation. Let, a = -3, b = 5, and c = -16. We can write (a × b) × c = 240 = a × (b × c). Please note that all binary operations are not associative, for example, subtraction denoted by '-'.

PROPERTY OF BINARY OPERATIONS

Commutative Property

A binary operation * on a non-empty set S is commutative, if a * b = b * a, for all (a, b) ∈ S. Suppose addition be the binary operation and N be the set of natural numbers. Let, a = 4 and b = 5, a + b = 9 = b + a.

PROPERTY OF BINARY OPERATIONS

Identity Property

A non-empty set P with a binary operation * is said to have an identity e ∈ P, if e*a = a*e= a, ∀ a ∈ P. Here, e is the identity element.

PROPERTY OF BINARY OPERATIONS

Inverse Property

A non-empty set P with a binary operation * is said to have an inverse element, if a * b = b * a = e, ∀ {a, b, e}∈P. Here, a is the inverse of b, b is the inverse of a and e is the identity element.

ELEMENTARY LOGIC
Oxford Dictionary defines mathematical logic as the part of mathematics concerned with the study of formal languages, formal reasoning, the nature of mathematical proof, provability of mathematical statements, computability, and other aspects of the foundations of mathematics. The following are terms unlocked concerning mathematical logic.

Logic - is the reasoning conducted or assessed according to strict principles of validity. It refers to the science that studies the principles of correct reasoning.
Axiom or Postulate - is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
Statement or Proposition - is a declarative sentence that is either true or false, but not both true and false.

Simple Statement - a statement that conveys a single idea.
Compound Statement - a statement that conveys a single idea.

NEGATION
The negation of a statement p, denoted by ~p (read as “not p”) is the statement whose truth value is the opposite of the truth value of p. Example:
p = Yellow is a color. ~p = Yellow is not a color.

CONJUNCTION
The conjunction of p and q, p ∧ q (read as “p and q”), is the statement that is true if and only if both p and q are true. In other words, if someone told you that he owned a dog and a cat, you would only consider that statement true if he really did own both animals, not just one or the other. Example:
p : 5 is a whole number.q : 5 is a positive integer.p ∧ q : 5 is a whole number and a positive integer.

DISJUNCTION
The disjunction of p and q, p ∨ q (read as “p or q”), is the statement that is true if and only if at least one of p and q is true. Disjunction has two types:

Inclusive or (∨) - or includes the possibility that both things may happen. Example

p : Troy is a singer.q : Troy is a teacher.p ∨ q : Troy is a singer or a teacher.

Exclusive or (⨁) - does not include the possibility that both may happen; one statement excludes the other. Note that in the truth table, when both statements are true and they use exclusive or, then p ⨁ q is false.

p : Troy is drinking water.q : Troy is singing.p ⨁ q : Troy is drinking water or singing.

CONDITIONAL
The conditional statement (also known as implication statement), p → q (read as “If p, then q”) is the statement that is true unless p is true and q is false. p is called the antecedent and q is called the consequent. If p → q is true, then sometimes p is called the hypothesis and q is called the conclusion. Example:
p : I buy a smartphone. (hypothesis)q : I get a free sim card. (conclusion)p → q: If I buy a smartphone then I get a free sim card.

BICONDITIONAL
The biconditional statement p ↔ q means (p → q) ∧ (p → q), or “if p then q, and if q then p.” We read p ↔ q as “p if and only if q”. Example:
p : Paul is a singer.q : Paul knows how to sing.p ↔ q: Paul is singer if and only if he knows how to sing.

ORDER OF PRECEDENCE
As a way of reducing the number of necessary parentheses, one may introduce precedence rules: ¬ (~) has higher precedence than ∧, ∧ higher than ∨, ∨ higher than →, and → higher than ↔. So for example, A ∨ B ∧ ¬ C → D, is short for (A ∨ (B∧(¬ C)) → D. Another example: Let p, q, and r represent the following.p : You get a promotion.; q : You complete the training.; r : You will receive a bonus.Answer: (p ∧ q ) → r If you get a promotion and you complete the training, then you will receive a bonus.

CONSTRUCTING A TRUTH TABLE
A truth table is a table that shows the truth value of a compound statement for allpossible truth values of its simple statements. When you're constructing a truth table, you have to consider all possible assignments of True (T) and False (F) to the component statements. For example, suppose the component statements are P, Q, and R. Each of these statements can be either true or false, so there are 2³= 8 possibilities. There are different ways of setting up truth tables. You can, for instance, write the truth values "under" the logical connectives of the compound statement, gradually building up to the column for the "primary" connective. The truth tables of the five operators discussed were already given above. See the example on the left. The first thing to do is analyze the logic pattern. As you can see, the negation was the first input followed by the conditional (since they are inside the parenthesis) and lastly the disjunction. Note that we should be extra careful in picking which columns are to be examined and partnered to be able to use the original truth tables.

QUANTIFIERS AND NEGATION
One way to create a statement from an open sentence is to substitute a specific element from the universal set for each variable in the open sentence. Another way is to make some claim about the truth set of the open sentence. This is often done by using a quantifier.

Existential quantifiers - are used as prefixes to assert the existence of something. Examples: some; there exists; at least one
Universal quantifiers - Examples: none, no = deny the existence of something; all, every = used to assert that every element of a given set satisfies some condition

Negating quantified statements is as easy as a piece of cake as long as you are familiar with the pattern of negation shown in a table in the figure on the right.

PROBLEM SOLVING AND REASONING
Inductive reasoning is the process of reaching a general conclusion by examining specific examples. The conclusion formed by using inductive reasoning is often called a conjecture, since it may or may not be correct. On the other hand, deductive reasoning is the process of reaching a conclusion by applying general assumptions, procedures, or principles. Here, conclusion is guaranteed unlike inductive reasoning.

INTUITION, PROOF, AND CERTAINTY

Intuition - The ability to understand something instinctively without the need for conscious reasoning or without having an actual evidence of it.

Example:Teacher: Square, rectangle and rhombus are parallelograms. A quadrilateral with opposite sides are parallel and equal. How many sides are there in a square?Student: Ma’am! Four.

Mathematical Proof - An argument that convinces other people that something is true.

Example: (Continuation)Teacher: How do you say so?Student: Ma’am, because quadrilateral means a polygon with four sides.

Certainty - A conclusion or outcome that is beyond doubt.

Example: (continuation)Teacher: Please show an illustration on the board.Student: (he draws square on the board)

RECREATIONAL PROBLEMS USING MATHEMATICS
This involves problems, riddles, puzzles, brain – teasers and games carried out for recreational or leisure rather than as a research or application-based professional activity. Though its main purpose is for recreation, it cannot be denied that the amount of mathematical content, understanding and procedure also requires some sort of rigor analysis and high degree of patience. Examples: Rubik’s cubes, Magic squares, fractals, logic puzzles and mathematical chess problems, ken-ken puzzles.

RUBIK'S CUBE

MAGIC SQUARE

LOGIC PUZZLE

KEN-KEN PUZZLE

POLYA'S FOUR STEPS PROBLEM SOLVING STRATEGY
According to George Polya (1887 - 1985), the four steps in solving problem are understanding the problem (look, visualize, organize, and connect the information given); devising a plan (make a representation, make a calculated guess, go through the process, or revise the problem); carrying out the plan (use mathematical knowledge and skills); and reviewing the solution (check and improve the solution if necessary).

Salazar_MAMWfinalexam_BSCE1B.pdf

MIDTERM EXAM (83/100)

REFLECTION
In this module, I was really overwhelmed with the pile of knowledge it contains. I think these were different lessons from different grade levels during my high school years. Yes, almost all of them were already discussed to us so I was able to review all of them. However, there were challenging parts especially those with a bit complex topics. Personally, I saw myself having hard time to understand and analyze the binary operations, some recreational problems, and constructing truth table. During the midterm exam, I struggled a lot in time-consuming problems such as the Polya's four steps problem solving and recreational problems specifically the logic puzzle. Meanwhile, I learned a lot in this module. Rest assured that the topics discussed will be used in the coming engineering problems that I will be solving. Thanks to the learning materials given to us as well as to our diligent teacher who discussed the topics in a way that everybody could understand.

MAMW Home

CLO3

CLO5

CLO1

CLO4

CLO6

REFERENCES:

https://i.pinimg.com/originals/67/93/26/6793268b1edd7e0f85b7fe4c6319811b.jpghttps://wallpaperaccess.com/full/3280804.jpghttps://res.cloudinary.com/jerrick/image/upload/v1581025680/5e3c898fc935ad001e25fc42.jpghttps://cutewallpaper.org/21/color-background/Free-photo-color-background-Abstract-Bright-Closeup-.pnghttps://i.pinimg.com/564x/24/b7/31/24b7317b0cbfb62e564ee191fa24e887.jpghttps://24wallpapers.com/app-gateway/wallpaper-uploads/wallpapers/legacyUploads/wi89124833a91-caf7-4c3e-aad5-42da99b7dba92079665946.jpghttps://media.istockphoto.com/photos/tropical-palm-leaf-texture-picture-id903532536?b=1&k=20&m=903532536&s=170667a&w=0&h=-vwffy-4G26nTv4uHDAdfcv-VBrUZ8FxsoBTJ-8iBVQ=https://www.igi-global.com/dictionary/mathematical-language/58744https://casioeducation.files.wordpress.com/2017/01/mathslang.jpghttps://i.ytimg.com/vi/xvwhKiEiHLs/maxresdefault.jpghttps://image.slidesharecdn.com/mathematicallanguageandsymbols-180624122316/95/mathematical-language-and-symbols-6-638.jpg?cb=1609840308https://study.com/cimages/multimages/16/889bc7ae-75b4-4681-95fc-6f3d3794a3c7_untitled.pnghttps://i.ytimg.com/vi/kvGsIo1TmsM/maxresdefault.jpghttps://www.katesmathlessons.com/uploads/1/6/1/0/1610286/different-ways-to-represent-a-function-mapping-table-graph-ordered-pairs_1_orig.pnghttps://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/relations-and-functions-1628659725.pnghttps://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/types-of-functions-1623849361.pnghttps://nool.ontariotechu.ca/mathematics/functions/intro-functions/evaluating-functions.phphttps://mathstats.uncg.edu/sites/pauli/112/HTML/section-56.htmlhttps://www.cuemath.com/algebra/binary-operation/https://images-na.ssl-images-amazon.com/images/I/31sD1hmwq7L.jpghttps://www.chilimath.com/wp-content/uploads/2020/03/truth-table-of-a-conjunction-702x1024.gifhttps://www.chilimath.com/wp-content/uploads/2020/03/truth-table-of-negation-850x1024.gifhttps://www.chilimath.com/wp-content/uploads/2020/03/truth-table-of-a-disjunction-665x1024.gifhttps://www.chilimath.com/wp-content/uploads/2020/03/truth-table-of-an-implication-739x1024.gifhttps://www.chilimath.com/wp-content/uploads/2020/03/truth-table-of-a-biconditional-919x1024.gifhttps://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.htmlhttps://i.ytimg.com/vi/19d6nq3hQGg/hqdefault.jpghttp://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/500px-Rubik%27s_cube.svg.pnghttps://upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Magicsquareexample.svg/1200px-Magicsquareexample.svg.pnghttps://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/KenKenSolution.svg/250px-KenKenSolution.svg.pnghttps://www.brainfans.com/brainteasers/brain-teaser-picture-logic-puzzles-mc-snack-corrected-4582126835e8b707aba1688.89816353.pnghttps://images.slideplayer.com/33/8236014/slides/slide_3.jpg

Page updated

Report abuse