GE003 - Module 2 - Mathematics Outline PDF

MODULE 2 OUTLINE Mathematical Language MODULE DURATION I. August 28 - 30, 2024 II. For asynchronous learning inquiries, you may reach me through messenger group/personal message from...

MODULE 2 OUTLINE Mathematical Language MODULE DURATION I. August 28 - 30, 2024 II. For asynchronous learning inquiries, you may reach me through messenger group/personal message from Monday to Friday 10 am – 5 pm LEARNING OBJECTIVES At the end of this module you are expected to: ⚫ Determine the four basic concepts of Mathematics as a language; ⚫ Apply set theory to organize and describe collections of objects; ⚫ Able to explain Mathematical Languages, symbols and conventions of mathematics by means of sets, set notations, set operation and functions; ⚫ Understand relations and functions and their significance in Mathematics Mathematics in the Modern World Module 2 MATHEMATICAL LANGUAGE The mathematical language is the system used to communicate mathematical ideas. This language consists of some natural language using technical terms (mathematical terms) and grammatical conventions that are uncommon to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas. The mathematical notation used for formulas has its own grammar and shared by mathematicians anywhere in the globe. CHARACTERISTICS OF MATHEMATICAL LANGUAGE The characteristics of mathematical language is being precise, concise and powerful. 1. Precision in mathematics is like a culture of being correct all the time. Definition and limits should be distinction. Mathematical ideas are being developed informally and being done more formally, with necessary and sufficient conditions stated up front and restricting the discussion to a particular class of objects. 2. Mathematical language must be concise or shows simplicity. Being concise is a strong part of the culture in mathematical language. The mathematician desires the simplest possible single exposition at the price of additional terminology and machinery to allow all the various particularities to be subsumed into the exposition at the highest possible level. 3. Mathematical language must also be powerful. It is a way of expressing complex thoughts with relative ease. The abstraction in mathematics is the desire to unify diverse instances under a single conceptual framework and allows easier penetration of the subject and the development of more powerful methods. EXPRESSION VERSUS SENTENCES An expression (or mathematical expression) is a finite combination of symbols that is well-defined according to rules that depend on the context. The symbols can designate numbers, variables, operations, functions, brackets, punctuations and groupings to help determine order of operations and other aspects of mathematical syntax. An expression is a correct arrangement of mathematical symbols used to represent the object of interest, it does not contain a complete thought, and it cannot be determine if it is true or false. Some of types of expressions are numbers, sets and functions. A sentence (or mathematical sentence) makes a statement about two expressions, either using numbers, variables or a combination of both. A mathematical sentence can also use symbols or words like equals, greater than, or less than. A mathematical sentence is a correct arrangement of mathematical symbols that states a complete thought and can be determined whether it is true, false, and sometimes true/sometimes false. CONVENTIONS IN THE MATHEMATICAL LANGUAGE Mathematical languages have conventions and it helps individuals distinguish between different types of mathematical expressions. Mathematical conventions is a fact, name, notation, or usage which is generally agreed upon by mathematicians. Mathematicians abide by conventions to be able to understand what they write without constantly having to redefine basic terms. Sets and Logic  Basic Operations and Relational Symbols  Set of Numbers FOUR BASIC CONCEPTS IN MATHEMATICS Language serves as a tool for teaching mathematical concepts. It can show how to make syntax and structure of mathematical language clear and explicit to understand the fundamental concepts. Language serves as a major pedagogical tool to understand how, what, and why things are said. LANGUAGE OF SETS Set Theory is the branch of mathematics that studies sets or the mathematical science of the infinite. The study of sets has become a fundamental theory in 1870. Introduced by Georg Cantor (German Mathematician.) SET – is a collection of a well-defined objects – usually denoted by capital letters of the alphabet and its members are enclosed with brackets. Elements – the members or objects of the set and is denoted by ∈. TWO WAYS OF REPRESENTING A SET A. Roster Method (Tabulation Method) – when the elements of the set are enumerated and separated by a comma. ex. Set D, E and F Example: a) Write the following in Roster Method. 1. A = {x/x is a positive integer less than 10. } Answer: A = {1, 2, 3, 4, 5, 6, 7, 8, 9} 2. B = {x/x is a month in the calendar. } Answer: B = {January, February,..., December} 3. C = {x/x is an integer, 1 < x < 8. } Answer: C = {2, 3, 4, 5, 6, 7} B. Rule Method (Set builder notation) – used to described the elements or members of the set. ex. Set A, B and C Example: b) Write the following in Rule Method. 1. D = {a, e, i, o, u} Answer: D = {x/x is a vowel} 2. E = {4 , 6 , 8 , 10, 12, 14 , 18, 20} Answer: E = {x/x is an even number between 3 and 21} 3. F = {12} Answer: F = {x/x represents a dozen} CARDINAL NUMBER – the number of members of the set. The cardinality of Set A is denoted by n(A). Example: A = {1,2,3,4,5,6,7,8,9} n(A) = 9 B = {x/x is a month in the calendar.} n(A) = 12 C = {x/x is an integer, 1 < x < 8. } n(A) = 6 D = {a, e, i, o, u} n(A) = 5 E = {4 , 6 , 8 , 10, 12, 14 , 18, 20} n(A) = 8 F = {12} n(A) = 1 TYPES OF SET 1. Finite Set – is a set whose elements are limited or countable and the last element can be identified. 2. Infinite Set – is set whose elements are unlimited or uncountable and the last element cannot be specified. 3. Unit Set – is a set with only one element, it is also called singleton. 4. Empty Set – a unique set with no elements and also called as the Null Set. It is denoted by {∅} or { }. 5. Universal Set – the totality of the set, all sets under investigation in any application of set theory are assumed to be contained in some large fixed set and is denoted by U Given: A = {a, b, c, } B = {a, b, c, d, e} C = {a, b, c, e, d} D = {f, g, h, i} E = {1, 2, 3, 4} F = {4, 5} U = {a, b, c, d, e, f, g, h, i, j, 1 , 2, 3, 4, 5} 6. Subset – If A and B are sets, A is called a subset of B, writtenA ⊆ B, if and only if, every element of A is also an element of B. A is a proper subset of B, written A ⊂ B, if and only if, every element of A is in B but there at least one element of B that is not in A. 7. Equal Set – two sets are equal if and only if, every element of A is in B and every element of B is in A. 8. Equivalent Set - two sets are equivalent if they have the same number of elements and is denoted by (∼). 9. Disjoint Set – if the two sets have no elements in common. Also called as non-intersecting set. OPERATIONS ON SETS 1. UNION OF SETS – The union of A and B, denoted by A ∪ B, is the set of all elements in x in U such that x is in A or x is in B. A = {a, b, c, } B = {c, d, e} C = {f, g} D = {f, g, h, i} a. A ∪ B = {a, b, c, d, e} b. C ∪ D = {f, g, h, i} c. B ∪ C = {c, d, e, f, g} *Note: Same element is written once in the final set. 2. INTERSECTION OF SETS – The intersection of A and B, denoted by A ∩ B, is the set of all elements in x in U such that x is in A and x is in B. A = {a, b, c, } B = {c, d, e} C = {f, g} D = {f, g, h, i} a. A ∩ B = {c} b. C ∩ D = {f, g} c. B ∩ C = { } 3. COMPLEMENT OF SET –The complement of A (or absolute complement of A), denoted by A’, is the set of all elements x in U such that x is not in A A = {a, b, c, } B = {c, d, e} U = {a, b, c, d, e, f, g, h} a. A′ = {d, e, f, g, h, } b. B′ = {a, b, f, g, h, } c. (A ∩ B)′ = {a, b, d, e, f, g, h} d. (A′ ∩ B′) = {f, g, h,} 4. DIFFERENCE OF SET – The difference of A and B (or relative complement of B with respect to A), denoted by A ∼ B, is the set of all elements x in U such that x is in A and x is not in B. A = {a, b, c, } B = {c, d, e} C = {f, g} D = {f, g, h, i} a. A − B = {a, b} b. C − D = { } c. B − C = {c, d, e} 5. CARTESIAN PRODUCT - The Cartesian product of sets A and B, written AxB, is AxB = {(a, b) | a ∈ A and b ∈ B} Example: Let A = {2, 3, 5} and B = {7, 8}. Find each set. a. A x B = { (2,7), (2,8), (3,7), (3,8), (5,7), (5,8)} b. B x A = { (7,2), (7,3), (7,5), (8,2), (8,3), (8,5)} c. A x A = { (2,2), (2,3), (2,5), (3,2), (3,3), (3,5), (5,2), (5,3), (5,5)} VENN DIAGRAM A Venn diagram is an illustration of the relationship between and among sets, groups of objects that share something in common. Usually, Venn diagrams are used to depict set intersections (denoted by an upside-down letter U). This type of diagram is used in scientific and engineering presentations, in theoretical mathematics , in computer applications and in statistics. LANGUAGE OF RELATIONS AND FUNCTIONS Relation and Function Definition Relation and function individually are defined as: Relations - A relation R from a non-empty A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.  Functions - A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image. Note: Please note that all functions are relations but all relations are not functions Representation of Relation and Function Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphical form, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, let us represent this function in different forms. Set-builder form - {(x, y): y = x2, x ∈ A, y ∈ B} Roster form - {(1, 1), (2, 4), (3, 9)} Arrow Representation – Difference Between Relation and Function The basic difference between a relation and a function is that in a relation, a single input may have multiple outputs. Whereas in a function, each input has a single output. The table given below highlights the differences between relations and functions. Note: Look at the example of the relation above: {(1, x), (1, y), (4, z)}. This is NOT a function, as a single element (1) is related to multiple elements (x and y). Hence the statement "every relation is a function" is incorrect. Terms Related to Relations and Functions Now that we have understood the meaning of relation and function, let us understand the meanings of a few terms related to relations and functions that will help to understand the concept in a better way: Cartesian Product - Given two non-empty sets P and Q, the cartesian product P × Q is the set of all ordered pairs of elements from P and Q, that is, P × Q = {(p, q) : p ∈ P, q ∈ Q} Domain - The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. It is called the set of inputs or pre-images. Range - The set of all second elements of the ordered pairs in a relation R from a set A to a set B is called the range of the relation R. It is called the set of outputs or images. Codomain - The whole set B in a relation R from a set A to a set B is called the codomain of the relation R. Note that range is a subset of codomain. i.e., Range ⊆ Codomain Types of Relations and Functions Different types of relations and functions have specific properties which make them different and unique. Let us go through the list of types of relations and functions given below: Types of Relations Given below is a list of different types of relations: Empty Relation - A relation is an empty relation if it has no elements, that is, no element of set A is mapped or linked to any element of A. It is denoted by R = ∅. Universal Relation - A relation R in a set A is a universal relation if each element of A is related to every element of A, i.e., R = A × A. It is called the full relation. Identity Relation - A relation R on A is said to be an identity relation if each element of A is related to itself, that is, R = {(a, a) : for all a ∈ A} Inverse Relation - Define R to be a relation from set P to set Q i.e., R ∈ P × Q. The relation R-1 is said to be an Inverse relation if R-1 from set Q to P is denoted by R-1 = {(q, p): (p, q) ∈ R}.  Reflexive Relation - A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R. Symmetric Relation - A binary relation R defined on a set A is said to be symmetric if and only if, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R. Transitive Relation - A relation R is transitive if and only if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for a, b, c ∈ A Equivalence Relation - A relation R defined on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Antisymmetric Relation -A relation R on a set A is said to be antisymmetric if (a, b) ∈ R and (b , a) ∈ R ⇒ a =b. LANGUAGE OF BINARY OPERATIONS The definition of binary operations states that "If S is a non-empty set, and * is said to be a binary operation on S, then it should satisfy the condition which says, if a ∈ S and b ∈ S, then a * b ∈ S, ∀ a, b ∈ S. In other words, * is a rule for any two elements in the set S where both the input values and the output value should belong to the set S. It is known as binary operations as it is performed on two elements of a set and binary means two. Properties of Binary Operation Let us learn about the properties of binary operation in this section. The binary operation properties are given below: 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. 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 '-'. 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. Distributive Property: Let * and # be two binary operations defined on a non-empty set S. The binary operations are distributive if, a* (b # c) = (a * b) # (a * c), for all {a, b, c} ∈ S. Suppose * is the multiplication operation and # is the subtraction operation defined on Z (set of integers). Let, a = 3, b = 4, and c = 7. Then, a*(b # c) = a × (b − c) = 3 × (4 − 7) = -9. And, (a * b) # (a * c) = (a × b) − (a × c) = (3 × 4) − (3 × 7) = 12 − 21 = -9. Therefore, a* (b # c) = (a * b) # (a * c), for all {a, b, c} ∈ Z. Identity Element: 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. 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. LEARNING ACTIVITY QUIZ: We will be having a Long Quiz next meeting. Topics covered from Nature of Mathematics until Mathematical Language and Symbol. RESOURCES: o https://www.mathsisfun.com/mathematics-language.html o https://www.docsity.com/en/mathematical-languages-and-symbols/4315119/ o https://www.onlinemathlearning.com/describing-sets.html o https://www.sciencedirect.com/science/article/pii/B9781483231235500069 o https://www.youtube.com/watch?v=c5ulafA3oe8 o https://www.math.tamu.edu/~Janice.Epstein/141/review/Chap6Review.pdf o https://www.purplemath.com/modules/setnotn.htm ohttps://www.google.com/search?q=venn+diagram&oq=venn&aqs=chrome.0.69i59j69i57j0l4j69i60l2.2662j0j7&sourc eid=chro me&ie=UTF-8 o https://www.mathsisfun.com/sets/venn-diagrams.html