Math 1100 Module 2: Mathematical Language & Sets PDF

Department of Mathematics and Physics MATH 1100 MODULE 21 Mathematical language and symbols Overview Mathematics has its own language which is used by mathematicians to communicate ideas, theories and concepts. Under...

Department of Mathematics and Physics MATH 1100 MODULE 21 Mathematical language and symbols Overview Mathematics has its own language which is used by mathematicians to communicate ideas, theories and concepts. Understanding its language and symbols is vital in learning mathematics. Knowing the language will help students understand and solve mathematical problems easier. In this module, students will be introduced to the basic mathematical language needed to express a range of mathematical concepts. Basic mathematical concepts such as sets, functions and relations, binary operations, and elementary logic will also be discussed. The main goal is to achieve quality education. Time allotment: 2 weeks Objectives: Upon completion of this module, you are expected to: 1. Discuss the language, symbol and conventions of mathematics. 2. Explain the nature of Mathematics as a language. 3. Perform operations on Mathematical expressions correctly. 4. Acknowledge that mathematics is a useful language. *These objectives are lifted from CHED’s CMO.48 (2017) PRE-ASSESSMENT 1. What do you think is the difference between expression and sentence in mathematical setting? 2. What are the two ways in writing or describing a set? 3. What are the five operations on sets? 4. Are all functions a relation? 5. What is a binary operation and what are its properties? 6. What are the logical connectives used in combining simple statements or propositions in logic? 1 This module is based from the book “Mathematics in the Modern World” by the Department of Mathematics and Physics, CS, CLSU. 1  Department of Mathematics and Physics MATH 1100 1. MATHEMATICS AS A LANGUAGE Language is important in person’s daily activities. People use language to create ideas and express them to other people. Similarly in the field of mathematics, mathematical language is used to express mathematical ideas and concepts. All language has their own vocabulary, and mathematics is not at exception. 1.1 Characteristics of mathematical language A. Precise Mathematical expressions or statements are precise, it has its own distinct meaning. Preciseness of mathematical expression or statements is best learned through understanding the language of mathematics. Example 1. Reducing the long English sentence, “The number of boys in a class, denoted by 𝑏, is less than the number of girls in a class, denoted by 𝑔.” symbolically into 𝑏 < 𝑔 greatly simplifies the sentence. The symbols retain the important and exact information and the context need only to be referred to again when stating a solution. B. Concise The language of mathematics is concise because it uses symbols instead of spelled- out words for shortness of statements. Example 2. The English sentence “Three plus eight equals eleven.” is expressed simply in the language of mathematics as 3 + 8 = 11. By the use of symbols, mathematical expressions become brief, and ambiguities are avoided. 2  Department of Mathematics and Physics MATH 1100 C. Powerful Mathematics is powerful because students can only perform well in problem solving if they understand the language of mathematics. To express mathematical ideas, students need to master particular requirements and conventions. In this way, complex ideas may be expressed in a greatly simplified manner. In other words, learning the language of mathematics empowers the students to be efficient problem solvers. Students also gain confidence in talking about their mathematical learning and articulate for themselves what else they need to learn. Overall, mathematical language skills include the abilities to read with comprehension, to express mathematical thoughts clearly, to reason logically, and to recognize and employ common patterns of mathematical thought. 1.2 Expressions vs. Sentences Mathematical language is composed of expressions and sentences. An expression is any correct arrangement of mathematics symbols, used to represent a mathematical object of interest; it does not state a complete thought and so it does not make sense to ask if it is true or false. Example 3. An example of expression can be as simple as 10 + 13. We could change the (+) to make different mathematical expressions such as 10 – 13, (10)(13), or 10 ÷ 13. Addition, subtraction, multiplication, and division are called operations. There are many more operations that can be used in a mathematical expression, which usually includes numbers, sets, functions, ordered pairs, matrices, and others. The following table lists the some key words used to express the four main operations. Mathematical Operations Addition Subtraction Multiplication Division Add Subtract Multiply Divide Increased by Decreased by Product Quotient 3  Department of Mathematics and Physics MATH 1100 Plus Minus Times Shared Sum Difference Twice Split between Total Reduced by Of Divided by More Less than Usually, verbal phrases are translated into variable expressions to simplify them into an equivalent form that usually involves fewer symbols and operations, or into a form that is best suited to a current application, or into a preferred form or style Some examples are shown below: Verbal Phrase Variable Expression The difference of 𝑥 and 𝑦 𝑥−𝑦 The sum of a number and ten 𝑥 + 10 A number increased by four 𝑥+4 Three more than a number 𝑥+3 Three less than a number 𝑥−3 A number minus three 𝑥−3 Six subtracted from a number 𝑥−6 Five times a number 5𝑥 Twice a number 2𝑥 12 The quotient of 12 and a number x 1 One half of a number x 2 The sum of five times a number and twelve 5𝑥 + 12 The product of six and twice a number (6)(2𝑥) The square of a number 𝑥2 The square root of a number √𝑥 The cube of a number 𝑥3 The sum of the cube of a number and four x3  4 The square of the sum of a number and eleven (𝑥 + 11)2 4  Department of Mathematics and Physics MATH 1100 Whenever possible, select a single variable to represent an unknown quantity. Then, express related quantities in terms of the selected variable. Look at the following examples. For each relationship, select a variable to represent one quantity and state what that variable represents. Then, express the second quantity in terms of the variable selected. 1. Two consecutive odd integers. Let 𝑥 = smaller odd integer 𝑥 + 2 = bigger odd integer If you are wondering why? First, list some consecutive odd integers: … , −5, −3, −1, 1, 3, 5, 7, 9, … How do you get the odd number next to the previous odd number? We add 2. 2. The tens digit of a two-digit number exceeds the units digit by 5. Let 𝑥 = units digit. 𝑥 + 5 = tens digit Here we are considering two quantities: the units digit and the tens digit of the two-digit number. How are these two quantities related? 3. The length of a rectangle is thrice its width. Let 𝑥 = width of the rectangle 3𝑥 = length of the rectangle We are considering two quantities: the length and the width of a rectangle. How are these two quantities related? 5  Department of Mathematics and Physics MATH 1100 A mathematical sentence is a correct arrangement of mathematical symbols stating a complete thought. The most common mathematical statements or sentences are called equations and inequalities. A mathematical sentence is one that makes a statement about the relationship of two expressions. These two expressions are written in symbols such as numbers and variables, or a combination of both. The relationship of the two expressions is usually stated by using symbols or words such as  equals (=),  greater than (>),  greater than or equal (≥),  less than ( 3𝑥 + 1 2. 2𝑥 − 18 = 6𝑦 + 1 3. A square has 𝑥 sides. In the above illustrations, the mathematical sentences may or may not be true depending on the values of the variables 𝑥 and 𝑦. The truth or falsity of such a sentence is open, depending on the values of the variables. On the other hand, we have a closed sentence if the mathematical sentence is definitely true or definitely false. Each of the following are closed sentences. Why? 1. The smallest prime number is 1. 2. The square root of 9 is 3. 3. 𝑥 − 2 = 𝑥 + 2 4. −112 is an even number. 5. 663 + 336 = 999 6. The square root of – 16 is – 4. 7  Department of Mathematics and Physics MATH 1100 2. BASIC MATHEMATICAL CONCEPTS Discussed below are fundamental concepts in mathematics; namely Sets, Functions and Relations, Binary Operations, and Logic. 2.1 Sets A set is a collection or grouping of elements. These elements can be anything such as numbers, letters, names, sentences etc. The capital letters 𝐴, 𝐵, 𝐶, … are usually used to name sets; if the elements are also letters, the small letters 𝑎, 𝑏, 𝑐, … are used. Describing sets  Roster (or List) Method  The simplest way of describing a set is to just list its elements separated by commas inside a pair of braces. It is easy to use especially if the set has only a few elements no matter what they are. Example: The set 𝐴 whose elements are 𝑎, 𝑏, and 𝑐 can be expressed as: 𝐴 = {𝑎, 𝑏, 𝑐} The order by which the elements are listed is irrelevant; a set is defined by what elements it contains, not by any ordering or priority among those elements. Thus, each of the following refers to the same set. 𝐴 = {𝑎, 𝑏, 𝑐 } 𝐴 = {𝑏, 𝑎, 𝑐 } 𝐴 = {𝑐, 𝑎, 𝑏} 𝐴 = {𝑎, 𝑐, 𝑏} 𝐴 = {𝑏, 𝑐, 𝑎} 𝐴 = {𝑐, 𝑏, 𝑎} Other examples: 𝐵 = {𝑚, 𝑎, 𝑡, ℎ} 𝐶 = {CAg, CASS, CBAA, CEd, Cen, CHSI, CF, CS, CVSM} 𝐷 = {1, 2, 3, 4, 5} 8  Department of Mathematics and Physics MATH 1100  Rule (or Description) Method  Another way of describing a set is giving a description that befits each of the elements. Example: 𝐴 = {𝑥 | 𝑥 is a vowel of the English alphabet}, which is read as “𝐴 is composed of any 𝑥 , where 𝑥 is a vowel of the English alphabet”. We call the number of elements of any set 𝐴 as the cardinal number of 𝐴. It is denoted as |𝐴|. Example: If 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ}, then |𝐴| = 8. If an element 𝑥 is a member of the set 𝐴, we write 𝑥 ∈ A; otherwise, we write 𝑥 ∉ 𝐴. Example: If 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ}, then 𝑓 ∈ 𝐴 but note that 𝐹 ∉ 𝐴. Definition: Set 𝐴 is a subset of set 𝐵, written 𝐴 ⊂ 𝐵, if every member of 𝐴 is also a member of 𝐵. Otherwise, we write 𝐴 ⊄ 𝐵, read “𝐴 is not a subset of 𝐵” to mean there is at least one element of 𝐴 that is not in 𝐵. Example: Given 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} and 𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒}. Then 𝐴 ⊂ 𝐵 because all members of 𝐴 are members of 𝐵. On the other hand, 𝐵 ⊄ 𝐴 because 𝑒 ∈ 𝐵 but 𝑒 ∉ 𝐵. It follows from the definition that any set 𝐴 is a subset of itself, i.e., 𝐴 ⊂ 𝐴. Note that ∈ and ⊂ mean two different concepts. To illustrate, 𝑎 ∈ {𝑎, 𝑏} but 𝑎 ⊄ {𝑎, 𝑏} {𝑎} ⊂ {𝑎, 𝑏} but {𝑎} ∉ {𝑎, 𝑏} {𝑎} ∈ {{𝑎}, 𝑏, 𝑐} but {𝑎} ⊄ {{𝑎}, 𝑏, 𝑐}. Definition: Two sets are equal, written 𝐴 = 𝐵, if and only if they have the same elements. Alternatively, 𝐴 = 𝐵, if and only if 𝐴 ⊂ 𝐵 and 𝐵 ⊂ 𝐴. 9  Department of Mathematics and Physics MATH 1100 Definition: Any set that has no element at all is called a null (or empty) set, denoted by { } or 𝜙. The null set is a subset of any other set. The set 𝐴 = { 𝑥 | 𝑥 is an integer between 1 and 2} is a null set. Definition: Any set that contains all elements under consideration is called a universal set, denoted by U. Whenever necessary in any discussion, the universal set is always given or identified. Operations On Sets Given a list of sets, other sets may be formed by performing one or more operations on the given sets. Basically these operations are the union (∪), intersection(∩), complement (′), difference (−), and the Cartesian or cross product (×).  Union The union (∪) operation combines all elements of two sets. Any element that occurs in both sets only occurs once in the new set. Example: If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑐, 𝑑, 𝑒} then 𝐴 ∪ 𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒}. If 𝐶 = {1, 2, 3, 5} and 𝐷 = {2, 4,6} then 𝐶 ∪ 𝐷 = ______________. (Answer2)  Intersection The intersection (∩) operation contains all elements found in two sets. In other words, the intersection of two sets contains only the elements common to both sets. Example: 1. If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑏, 𝑐, 𝑑, 𝑒} then a. 𝐴 ∩ 𝐵 = {𝑏, 𝑐} b. (𝐴 ∩ 𝐵) ∪ 𝐵 = {𝑐, 𝑑, 𝑒} 2. If 𝐶 = {1, 2, 3, 5} and 𝐷 = {2, 4,6} then 𝐶 ∩ 𝐷 = ______________. (Answer3) 3. If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐶 = {1, 2, 3, 5} then 𝐴 ∩ 𝐶 = { }. 2 𝐶 ∪ 𝐷 = {1, 2, 3,4,5,6} 3 𝐶 ∩ 𝐷 = {2} 10  Department of Mathematics and Physics MATH 1100  Complement The complement (′) of a set, denoted 𝐴’, identifies the elements of the universal set 𝑈 that are not in 𝐴. Examples: 1. If 𝐴 = {𝑥, 𝑦} and 𝑈 = {𝑥, 𝑦, 𝑧}, then 𝐴’ = {𝑧}. 2. If 𝐶 = {2, 4, 6, 8} and 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9} then 𝐶 ′ = _______. (Answer4) 3. Ø’ = 𝑈  Difference The difference of two sets 𝐴 and 𝐵, denoted 𝐴 − 𝐵, is defined to be the set whose elements are those of 𝐴 that are not in 𝐵. Example: If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑐, 𝑑, 𝑒}, then a. 𝐴 − 𝐵 = {𝑎, 𝑏} b. 𝐵 – 𝐴 = {𝑑, 𝑒}. For the complement and difference operations, 𝐴′ = 𝑈 − 𝐴 𝐴 − 𝐵 = 𝐴 ∩ 𝐵′  Cartesian Product The Cartesian Product or Cross Product of two sets 𝐴 and 𝐵, denoted 𝐴 × 𝐵, is the set of all ordered pairs (𝑥, 𝑦), such that 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵. Example: If 𝐴 = {𝑎, 𝑏} and 𝐵 = {1, 2, 3}, then a. 𝐴 × 𝐵 = {(𝑎, 1), (𝑎, 2), (𝑎, 3), (𝑏, 1), (𝑏, 2), (𝑏, 3)} b. 𝐵 × 𝐴 = {(1, 𝑎), (1, 𝑏), (2, 𝑎), (2, 𝑏), (3, 𝑎), (3, 𝑏)} Note: (3, 𝑎) ∉ 𝐴 × 𝐵 SUPPLEMENTARY VIDEOS: For better understanding of the topics on sets, video links are provided below. Iba't ibang Pamamaraan sa Pagsulat ng Set Notation Algebra - Basic Set Notation - YouTube Paano Makuha ang Subset ng isang Given Set - YouTube 4 𝐶′ = {1, 3, 5, 7, 9} 11