Mathematical Language and Symbols PDF

Mathematical Language and Symbols Language is the system of words, signs and symbols which people use to express ideas, thoughts and feelings. Language consists of the words, their pronunciation and the methods of combining them to be understood by a community. Language is a systematic means...

Mathematical Language and Symbols Language is the system of words, signs and symbols which people use to express ideas, thoughts and feelings. Language consists of the words, their pronunciation and the methods of combining them to be understood by a community. Language is a systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures or marks having understood meanings Language is a systematic means of communicating by the use of sounds or conventional symbols. Language is a system of words used in a particular discipline. Language is a set (finite or infinite) of sentences, each finite in length and constructed out of a finite of elements. The definition describes the language in terms of the following components: - A vocabulary of symbols or words. - A grammar consisting of rules of how these symbols may be used. - A syntax or proportional structure, which places the symbols in linear structures. - A discourse or narrative consisting of strings of syntactic propositions. - A community of people who use and understand these symbols. - A range of meaning that can be communicated with these symbols. Mathematics in the English-speaking world is communicated using two Languages 1. Mathematical English - It is part of the English language used for making formal mathematical statements, specifically to communicate definitions, theorems, proofs and examples. Many ordinary English words are used in Mathematical English with different meanings. In some ways, Mathematical English is a foreign language. Other languages also have special mathematical forms. Mathematics in the English-speaking world is communicated using two Languages 2. Symbolic Language - The symbolic language of mathematics is a special- purpose language. It has its own symbols and rules of grammar that are quite different from those of English. We can usually read expressions in the symbolic language in any Mathematical article written in any language. The vocabulary of Mathematics The symbolic language consists of symbolic expressions written in the way mathematicians traditionally write them. A symbol is a typographical character such as 𝑥, Φ, ∪, ∞. It is also includes symbols that are specific to mathematics, such as 0 is a symbolic assertion. b. 𝑥 > 0 is true for 𝑥 = 5 and many other numbers and false for 𝑥 = −0.001 and many other numbers. c. The symbolic assertion 𝑥 2 − 4𝑥 + 1 = 0 is true for the number 𝑥 = 2, but not for any other number. d. 𝑥 2 − 𝑦 2 = (𝑥 − 𝑦)(𝑥 + 𝑦) is a symbolic assertion with two variables. Types of Symbolic Expressions 2. Symbolic Statements - A symbolic assertion without variables. It is either true or false. Example: a. 𝜋 > 0 and 42 = 16 are symbolic statements. b.𝜋 < 0 and 2 + 3 = 7 are false symbolic statements. Even though false, they are still regraded as symbolic statements. Types of Symbolic Expressions 3. Symbolic Terms - A symbolic expression that refers to some mathematical object. Example: a. The expression 52 is a symbolic term. It is another name for the number 25. b. 𝑥 3 is a symbolic term containing a variable 𝑥. This means the term containing a variable 𝑥. This means the term has variable meaning depending on which value is substituted for 𝑥. For example, if you set 𝑥 = 2, we get 23 , another name for 8. c. 𝑥 2 + 𝑦 2 − 𝑥𝑦 is symbolic term with two variables. If you substitute 2 for 3 for y then the expression denotes the integer 7. Grammar of the Symbolic Language - The symbolic language of mathematics has its own rules of grammar that are quite different from those of English. 1. In symbolic expressions, the symbols and the arrangement of the symbols both communicate meaning: Examples: a. 𝑐𝑜𝑠2𝑥, 2𝑐𝑜𝑠𝑥, and cos2 𝑥 all mean different things. b. sin2 𝑥 and 𝑠𝑖𝑛𝑥 2 mean the same thing. c. 𝑥2𝑐𝑜𝑠 is meaningless. 2. In mathematics the order of operation is more important. It is a collection of some rules which gives the procedures to perform first in order to evaluate a given mathematical expressions. Examples a. 2 ∙ 5 + 3 means the multiplication first, then add the five, getting 13, whereas 2 ∙ (5 + 3) means do the addition first, then multiply the result by 2, getting 16. b. 4 + 32 first calculate 32 = 9 getting 4 + 9, then calculate 4 + 9, getting 13. But 4 + 3 2 means 72. Mathematical Language is the system used to communicate mathematical ideas. Mathematical Language has its own grammar, syntax, vocabulary, word order, synonyms, conventions, idioms, abbreviations, sentence structure and paragraph structure. It has certain language features unparalleled in other languages, such as representation. Mathematical Language also includes a large component of logic. The ordinary language which gradually expands to comprise symbolisms and logic leads to learning of mathematics and its useful application to problem situations. Four main actions attributed to problem-solving and reasoning 1. Modelling and Formulating: Creating appropriate representations and relationships to mathematize the original problem. 2. Transforming and manipulating: Changing the mathematical form in which a problem is originally expressed to equivalent forms that represent solutions. 3. Inferring: Applying derived results to the original problem situation, and interpreting and generalizing the results in that light. 4. Communicating: Reporting what has been learned about a problem to a specified audience. Objectives The students will be able to: 1. Discuss the language, symbols, and conventions of mathematics; 2. Explain the nature of mathematics as a language; 3. Perform operations on mathematical expressions correctly; and 4. Acknowledge that mathematics is a useful language. Characteristics of Mathematical Language The use of language in mathematics differs from the language of ordinary speech in three important ways: 1. Mathematical language is non-temporal. There is no past, present or future in mathematics. 2. Mathematical language is devoid of emotional content. 3. Mathematical language is precise. Vocabulary understanding is a major contributor to overall comprehension in many content areas, including mathematics. Operation Terms and Symbols Addition Subtraction Multiplication Division + − 𝑥, ,∗ [÷,/] - Plus - Minus - Multiplied by - Divided by - The sum of - The difference - The product of - The quotient of - Increased by of - Times of - per - Total - Decreased by - Subtracted from Multivariate mathematical expressions have more than one variables: Example: - 5𝑥𝑦 + 9𝑥 − 12 - 31𝑎𝑏𝑐 - 9𝑦/3𝑥 Mathematical Expressions consist of terms. The terms is separated from other terms with either plus or minus signs. A single term may contain an expression in parentheses or other grouping symbols. Types of Mathematical Expressions  Mathematical Sentence combines two mathematical expressions using a comparison operator. These expressions either use numbers, variables or both. The comparison operators include equal, not equal, greater than, greater than or equal to, less than and less than or equal to. The signs which convey equality or inequality are also called relation symbols because they specify how two expres-sions are related. A mathematical expressions containing the equal sign is an equation. The two parts of an equation are called members. A mathematical expression containing the inequality sign is an inequality. Example of an equation Example of Inequalities Conventions in the Mathematical Language 2 things to consider to understand symbols Context – refers to the particular topics being studied and it is important to understand the context to understand mathematical symbols. Convention – is a technique used by mathematicians, engineers, scientists in which each particular symbol has particular meaning. Greek and Latin letters are used as symbols for physical quantities and special functions and conventionally, for variables representing certain quantities. Basic Concepts 1. SETS Definition. A set is a well- defined collection of distinct objects. The objects that make up a set is called elements.  Two ways to describe set 1. Roster/Tabular 2. Rule/Descriptive The elements in the given set are listed or The common characteristics of the elements are enumerated, separated by a comma, inside defined. This method uses set builder notation a pair of braces where x is used to represent any element of the given set.  Kinds of Sets 1. Empty/Null/Void Set has no element and is denoted by ∅ by a pair of braces with no element inside, i.e. {} 2. Finite Set has countable number of elements, i.e. A = {1, 2, 3, 4, 5} 3. Infinite Set has uncountable number of elements, A = {…, -3, -2, -1, 0, …} 4. Universal Set is the totality of all the elements of the sets under consideration, denoted by U, i.e. U = {…-2, -1, 0, 1, 2,…} Two or more sets may be related to each other as described by the ff: 1. Equal Sets have been the same elements 2. Equivalent Sets have the same number of elements 3. Joint Sets have at least one common element 4. Disjoint Sets have no common element Subset is a set every element of which can be found on a bigger set. The symbol ⊂ means “a subset of” while ⊄ means “not a subset of”. Improper Subset (⊆) – if the first set equals the second set. A null set is always a subset of any given set and is considered an improper subset of the given set. Proper Subset (⊂) – other than the set itself and the null set, all are considered proper subsets. Power Set – the set containing all the subsets of the given set with n number of elements. Operations of Sets Suppose there are Sets A and B. 1. Union of Sets A and B [denoted by A ∪ 𝐵] is a set whose elements are found in A and B or in both. In symbol: 𝐴 ∪ 𝐵 = {𝑥/𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}. 2. Intersection of Sets A and B [denoted by 𝐴 ∩ 𝐵] is a set whose elements are common to both sets. In symbol: 𝐴 ∩ 𝐵 = {𝑥/𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}. 3. Difference of Sets A and B [denoted by 𝐴 − 𝐵] is a set whose elements are found in set A but not in set B. In symbol: 𝐴 − 𝐵 = {𝑥/𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵}. 4. Complement of Set A [denoted by A’] is a set whose elements are found in the universal set but not in set A. In symbol: A’ = {𝑥/𝑥 ∈∪ 𝑎𝑛𝑑 𝑥 ∉ 𝐴}. The pictorial representation of relationship and operations of set is the so called Venn- Euler Diagrams or simply Venn Diagrams. The universal set is usually represented by a rectangle while circles with the rectangle usually represent it subsets. The shaded region in the diagrams illustrates the sets relation or operation.  Leonhard Euler (15 April 1707 – 18 John Venn (4 August 1834 – 4 April 1923) - September 1783) – was a Swiss an English Logician an Philosopher mathematician, physicist, astronomer, logician, and engineer. 2. FUNCTIONS Definition. Functions are mathematical entities that give unique outputs to particular inputs. A functions consists of argument (input to a function), value (output), domain (set of all permitted inputs to given function) and codomain (set of permissible outputs). Operations of Functions Let f and g are the given functions: 1. The sum f + g is the function defined by: y = (f + g) x = f(x) + g(x) 2. The difference of f – g is the function by: y = (f – g) x = f(x) – g(x) 3. The product f*g is the function defined by: y = (f*g) x = f(x)*g(x) 𝒇 𝑓 𝑓 𝑥 4. The quotient is the function defined by: y = 𝑥= 𝒈 𝑔 𝑔 𝑥 Example Given: 𝑓 𝑥 = 3𝑥 + 2, 𝑔 𝑥 = 4 − 5𝑥 Find: 𝑓 2 + 𝑔 3 Solution: 𝑓 2 +𝑔 3 =3 2 +2+ 4−5 3 𝑓 2 +𝑔 3 = 8 + 4 − 15 𝑓 2 +𝑔 3 = 8 + −11 𝑓 2 +𝑔 3 = 8 − 11 𝑓 2 +𝑔 3 =3 Given: 𝑓 𝑥 = 3𝑥 + 2, 𝑔 𝑥 = 4 − 5𝑥 Find: 𝑓 𝑥 + 1 − 𝑔 3 Solution: 𝑓 𝑥+1 +𝑔 3 = (3 𝑥 + 1 + 2) − 4 − 5 3 𝑓 𝑥+1 +𝑔 3 = 3𝑥 + 3 + 2) − 4 − 15 𝑓 𝑥+1 +𝑔 3 = (3𝑥 + 3 + 2 − −11 ) 𝑓 𝑥+1 +𝑔 3 = (3𝑥 + 3 + 2 + 11) 𝑓 𝑥+1 +𝑔 3 = 3𝑥 + 16 Given: 𝑓 𝑥 = 3𝑥 + 2, 𝑔 𝑥 = 4 − 5𝑥 𝑓 2 Find: 𝑔 3 𝑓 2 3 2 +2 6+2 8 8 Solution: = = = = − 𝑔 3 4−5(3) 4−15 −11 11 Given: 𝑓 𝑥 = 3𝑥 + 2, 𝑔 𝑥 = 4 − 5𝑥 𝑓 𝑥 Find: 𝑔 𝑥 𝑓 𝑥 3𝑥+2 Solution: = 𝑔 𝑥 4−5𝑥 Given: 𝑓 𝑥 = 𝑥 + 2, 𝑔 𝑥 = 𝑥 2 − 4 𝑓 𝑥 Find: 𝑔 𝑥 𝑓 𝑥 𝑥+2 𝑥+2 1 Solution: Find: = = = 𝑔 𝑥 𝑥 2 −4 (𝑥−2)(𝑥+2) 𝑥−2 Given: 𝑓 𝑥 = 𝑥 + 2, 𝑔 𝑥 = 𝑥 2 − 4 Find: 𝑓 𝑥 ∗ 𝑔(𝑥) Solution: 𝑓 𝑥 ∗ 𝑔 𝑥 = 𝑥 + 2 𝑥2 − 4 𝑓 𝑥 ∗ 𝑔 𝑥 = (𝑥 3 − 4𝑥 + 2𝑥 2 − 8) 𝑓 𝑥 ∗ 𝑔 𝑥 = 𝑥 3 + 2𝑥 2 − 4𝑥 − 8 Composite Function: 𝑓 𝑔 𝑥 = 𝑓°𝑔 𝑥+1 Example. Given: 𝑓 𝑥 = 2𝑥 − 𝑥, 𝑔 𝑥 = 2 Find 𝑓 𝑔 𝑥 𝑥+1 𝑥+1 𝑓 𝑔 𝑥 =2 − 2 2 𝑥+1 𝑓 𝑔 𝑥 = (𝑥 + 1) − 2 2 𝑥+1 − 𝑥+1 𝑓 𝑔 𝑥 = 2 2𝑥 + 2 − 𝑥 − 1 𝑥 + 1 𝑓 𝑔 𝑥 = = 2 2 𝑥+1 𝑓 𝑔 𝑥 = 2 3. RELATIONS Definition. A relation is a set of inputs and outputs, oftentimes expressed as ordered pairs (input, output). A relation is a rule which associates each element of the first set with at least one element in the second set.  Example When an independent variable corresponds to more than one variable, it is a relation. A relation is a correspondence between a first set of variable such that for some elements of the first set of variables, there correspond at least two elements of the second set of variables. 4. BINARY OPERATIONS Definition. Binary means consisting of two parts. In mathematics, binary means that it belongs to a number system with base 2 and not base 10. A binary is made up of 0’s and 1’s. A bit is a single binary digit. Transform the binary number 1111112 to decimal. 1 0 0 0 0 0 0 = 26 = 64 1 0 0 0 0 0 = 25 = 32 1 0 0 0 0 = 24 = 16 1 0 0 0 = 23 = 8 1 0 0 = 22 = 4 1 0 = 21 = 2 1 = 20 = 1 1 1 1 1 1 1 1 = 127 Since 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127 Convert the decimal number to binary number 1. 2510 = 110012 =000110012 =198 20 21 22 23 24 1 2 4 8 16 1 0 0 1 1 25 – 16 = 9 9–8=1 Convert the decimal number to binary number 1. 80.2510 = 1010000.012 Left side before decimal point 20 21 22 23 24 25 26 1 2 4 8 16 32 64 0 0 0 0 1 0 1 80 – 64 = 16 For right: 0.25 x 2 = 0.5 0 0.5 x 2 = 1.0 1 Therefore, 0.25 = 012 Converting Binary to Decimal Given: 1100112 = 1 1 0 0 1 1 25 24 23 22 21 20 32 16 8 4 2 1 32 + 16 + 0 + 0+ 2+ 1= 51 Converting Binary to Decimal Given: 110010.012 = 1 1 0 0 1 0 0 1 25 24 23 22 21 20 2−1 2−2 32 16 8 4 2 1 0.5 0.25 32 + 16 + 0 + 0+ 2+ 0 + 0 + 0.25 = 50.25 To convert a decimal number to binary number: Decimal Binary 0 0 1 20 12 2 21 + 0 102 3 21 + 1 112 4 21 + 0 + 0 1002 5 21 + 0 + 1 1012 6 22 + 21 + 0 1102 Binary Operation is a rule of combining two values to produce a new value. A binary operations is said to be commutative if the order of the arguments is charged and the result is equivalent. A binary operation is said to be associative if the order of the parentheses is changed and the result is equivalent. An element, denoted by e, is said to be identity or neutral element of the binary operation if under the operation, any element combined with e results in the same element. For an element, the inverse represented as a -1, when combined with under the binary operation results in the identity element for that binary operation. Typical examples of binary operations are the addition and multiplication of numbers and matrices, as well as composition of functions on a single set. Examples: On the set of real numbers R, f(a, b) = a + b is a binary operation since the sum of two real numbers is a real number. On the set natural numbers N, f(a, b) = a + b is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different. On the set M(2, 2) of 2 x 2 matrices with real entries, f(A, B) = AB is a binary operation since the sum of two such matrices is another 2 x 2 matrix. On the set M (2,2) of 2 x 2 matrices with real entries, f(A, B) = A + B is a binary operation since the product of two such matrices is another 2 x 2 matrix. For a given set C, let S be the set of all functions h: C → C. Define f: S x S → S by 𝑓(ℎ1 , ℎ2 )(𝑐) = (ℎ1 , ℎ2 )(𝑐) = ℎ1 (ℎ2 (𝑐)) for all 𝑐 ∈ 𝐶, the composition of the two functions ℎ1 and ℎ2 in S. Then f is a binary operation since the composition of the two functions is another function on the set C (that is, a member of S). Elementary Logic Definition. Logic is the science of formal principles of reasoning or correct inference. It is the study of principles and methods used to distinguish valid arguments from those that are not valid. Logic is the expressions of ordered thoughts starting from axioms and resulting in a conclusion. Additional Information. Mathematical Logic is the study of reasoning in mathematics. Mathematical Reasoning is deductive; meaning it consists of drawing conclusions from given hypothesis. Note: A more detailed discussion about logic will be presented in Chapter V Formality Formality is a relational concept: an expression can be more or less formal relative to another expression, entailing an ordering of expressions; yet, no expression can be absolutely formal or absolutely informal