Mathematics in the Modern World PDF

Mathematics in the Modern World Chapter 2: Mathematical Language and Symbols 2.1 Mathematics as a Language  What is language? Language is a systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures, or marks having understood meanings....

Mathematics in the Modern World Chapter 2: Mathematical Language and Symbols 2.1 Mathematics as a Language  What is language? Language is a systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures, or marks having understood meanings. Merriam-Webster dictionary  According to Dr. Burns, “the language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is: 1.precise (able to make very fine distinctions); 2.concise (able to say things brief); 3.powerful (able to express complex thoughts with relative ease).”  Some Classification of Symbols 1. Numbers A number is a mathematical object used to count, quantify, and label another object. These include the elements of the set of real numbers (ℝ), rational numbers (ℚ), irrational numbers (ℚ’), integers (ℤ), and natural numbers (ℕ). Some Classification of Symbols 2. Operation Symbols include addition (+), subtraction (-), multiplication (x or ), division ( or /) , and exponentiation (𝑥 𝑛 ), where x is the base and n is the exponent.  Some Classification of Symbols 3. Relation Symbols include greater than or equal ( ) , less than or equal (), equal ( ), not equal ( ), similar (), approximately equal (), and congruent (). Congruent figures are the same shape and size. Similar figures are the same shape, but not necessarily the same size. On the other hand, two quantities are approximately equal when they are close enough in value so the difference is insignificant in practical terms. Some Classification of Symbols 4. Grouping Symbols include parentheses ( ), curly brackets or braces { }, or square brackets [ ]. 5. Variables are another form of mathematical symbol. These are used when quantities take different values. These usually include letters of the alphabet.  Some Classification of Symbols 6. Set theory symbols these are those used in the study of sets. These include subset (  ), union (), intersection (), element (), not element (), and empty set (  ). 7. Logic symbols include implies (), equivalent (), and (), or (), for all (), there exists (), and therefore (). Some Classification of Symbols 8. Statistical symbols include sample mean (𝑥), population mean (), median ( 𝑥 ), population standard deviation (), summation (  ) and factorial (n!), among others. Mathematical Expression and Mathematical Sentence A mathematical expression (analog of a ‘noun’) defined as a mathematical phrase that comprises a combination of symbols that can designate numbers (constants), variables, operations, symbols of grouping and other punctuation. However, this does not state a complete thought. Mathematical Expression and Mathematical Sentence A mathematical sentence makes a statement about two expressions. The two expressions either use numbers, variables, or a combination of both. It uses symbols or words like equals, greater than, or less than and it states a complete thought. Types of Sentences An open sentence is a sentence that uses variables; thus it is not known whether or not the mathematical sentence is true or false. A closed sentence, on the other hand, is a mathematical sentence that is known to be either true or false. Example The following are mathematical sentences. Label each of the following as open or closed. For those closed sentences, identify if it is true or false. 1. 10 is an odd number. Answer: Closed - false 2. 4 + 5x = 9 Answer: Open 3. 10 - 1 = 7 + 2 Answer: Closed - true 4. 6 - x = 5 Answer: Open 5. The square root of 4x is 2. Answer: Open Translating Phrases to Mathematical Expressions or Sentences Multiplication Addition (+) Subtraction (−) Division (÷) (×) minus twice (times 2) combined with the difference of thrice (times 3) divided by plus decreased by squared the quotient of the sum of fewer than cubed half of increased by less than times a third of total subtracted from the product of ratio more than less multiplied by shared equally added to take away of Translating Phrases to Mathematical Expressions or Sentences Less than or Greater than or Equal ( = ) equal (  ) equal (  ) Equals Is Is the same at most at least as not greater than not less than Yields amount to Example Two times a number increased by 4 is 14. Answer: 2n + 4 = 14 2n +4 = 14 Ten more than thrice a number is at least 12. Answer: 3n + 10  12 The sum of two consecutive integers is 25. Answer: n + (n +1) = 25 Subtract 3x from 10xy. Answer: 10xy - 3x Ten more than four times a number less than six. Answer: 6 – (4x + 10) Ten more than four times a number is less than six. Answer: 4x + 10 < 6 Nine less a number n Answer: 9 - n 2.2 Four Basic Concepts: Set, Relation, Function and Binary Operation  Sets A set is a well-defined collection of distinct objects. The objects in sets can be anything: numbers, letters, movies, people, animals, etc. Each object belonging to a set is called the element or member of the set. For example, the set 𝐶 of counting numbers less than 4 has numbers 1, 2 and 3 as the elements. We use the notation “ ∈ ” to indicate that a specific element belongs to a set; otherwise, we use “∉”. Thus, we write 1 ∈ 𝐶 and 0 ∉ 𝐶 to mean that 1 is an element of 𝐶 and 0 is not an element of 𝐶, respectively.  There are two ways of specifying a set, namely, roster method and rule method. In the roster method, the elements of the set are enumerated, separated by a comma (,), and enclosed in a pair of braces ({ }). In the rule method, a phrase is used to describe all the elements in the set. Definition of terms The set with no elements is called the empty set or null set and is denoted by ∅ or { }. The set with only one element is called the singleton set. If a set contains all the elements under consideration, then it is called a universal set, denoted by 𝑼. A set is finite if it consists of a finite number of elements; otherwise, it is infinite. Definition of terms Two sets, say 𝐴 and 𝐵, are said to be equal, written 𝐴 = 𝐵, if 𝐴 and 𝐵 have exactly the same elements. If 𝐴 and 𝐵 have the same number of elements, then we say that 𝐴 and 𝐵 are equivalent sets. A set 𝐴 is called a subset of a set 𝐵, written 𝐴 ⊆ 𝐵, if and only if every element of 𝐴 is also an element of 𝐵. If 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then we say that 𝐴 is a proper subset of 𝐵, and write 𝐴 ⊂ 𝐵. Remarks i. The null set is a proper subset of every set. ii.Any set is a subset of itself. iii.A set with 𝑛 elements has a total of 2𝑛 subsets. Operations on Sets Let 𝐴 and 𝐵 be two arbitrary sets. The union of 𝐴 and 𝐵, written 𝐴 ∪ 𝐵, is the set containing all the elements which belong to either 𝐴 or 𝐵 or to both. The intersection of 𝐴 and 𝐵, written 𝐴 𝐵 , is the set of all elements which are common to both 𝐴 and 𝐵. The complement of a set 𝐴, written 𝐴′ , is the set of all elements which are in the universal set 𝑈 but not in 𝐴. Example Consider the sets 𝑈 = {1, 2, 3, 4, 5}, 𝐴 = 1, 5 , and 𝐵 = {2, 3, 5}. Then 1. 𝐴 ∪ 𝐵 = 1, 5 ∪ 2, 3, 5 = {1, 2, 3, 5} 2. 𝐴 ∩ 𝐵 = 1, 5 ∩ 2, 3, 5 = {5} 3. 𝐵′ = 2, 3, 5 ′ = {1, 4} 4. 𝐴 ∪ 𝐵′ = 1,5 ∪ 1,4 = {1, 4, 5}  Let 𝐴 and 𝐵 be two non-empty sets. The Cartesian product of sets 𝐴 and 𝐵, denoted by 𝐴 × 𝐵, is the set of all ordered pairs (𝑎, 𝑏) where 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵. Example Consider again the sets in the previous example, 𝑈 = {1, 2, 3,4, 5}, 𝐴 = 1, 5 , and 𝐵 = {2, 3, 5}. We have 1. 𝐴 × 𝐵 = { 1,2 , 1,3 , 1,5 , 5,2 , 5,3 , (5,5)} 2. 𝐴 × 𝐴 = { 1,1 , 1,5 , 5,1 , 5,5 } 3. 𝐵 × 𝐴 = { 2,1 , 2,5 , 3,1 , 3,5 , 5,1 , 5,5 } Relations and Functions Intuitively, a ‘relation’ is just a relationship between sets of information. The couple pairing and the pairing of students’ names and the courses taken are examples of a relation. In mathematics, a relation 𝑅 from set 𝑋 to set 𝑌 is a subset of 𝑋 × 𝑌. If (𝑥, 𝑦) ∈ 𝑅, then we say that 𝑥 is related to 𝑦 (or 𝑦 is in relation with 𝑥). Illustration: function 𝒇  The domain of the relation 𝑅, denoted by 𝐷(𝑅), is the set of all first coordinates in the ordered pairs which belong to 𝑅. That is, 𝐷 𝑅 = 𝑥: 𝑥 ∈ 𝑋, 𝑥, 𝑦 ∈ 𝑅. The image of the relation 𝑅 , denoted by 𝐼(𝑅), is the set of all second coordinates in the ordered pairs in 𝑅. That is, 𝐼 𝑅 = 𝑦: 𝑦 ∈ 𝑌, 𝑥, 𝑦 ∈ 𝑅.  𝑿 𝒀 𝒇 𝑥1 𝑦1 𝑥2 𝑦2 𝑥3 𝑦3 Fig 1. Mapping diagram of 𝑓: 𝑋 → 𝑌 Binary Operations Let 𝑆 be a non-empty set. A binary operation ∗ on 𝑆 is a function from 𝑆 × 𝑆 into 𝑆 such that for 𝑥, 𝑦 ∈ 𝑆, we have 𝑥 ∗ 𝑦 for ∗ (𝑥, 𝑦). Note that the image of ∗ is a subset of 𝑆. Thus, we say that 𝑆 is closed under ∗. Example 1.The usual addition (+) , subtraction (−) and multiplication (∙) are binary operations on the set ℝ of real numbers. 2.Subtraction (−) and division (÷) are not binary operations on the set ℕ since 1 − 2 ∉ ℕ and 2 ÷ 3 ∉ ℕ. 3.Let 𝑃 be the set of all sets. The union ∪ and intersection ∩ of sets are binary operations on 𝑃. Properties of Binary Operations 1.Commutative property A binary operation is commutative, if ∀ 𝑥, 𝑦 ∈ 𝑆, 𝑥 ∗ 𝑦 = 𝑦 ∗ 𝑥. 2.Associative property A binary operation * on 𝑆 is associative, if ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑆, 𝑥 ∗ 𝑦 ∗ 𝑧 = 𝑥 ∗ 𝑦 ∗ 𝑧.