02_Handout_1(9) MMW.pdf

GE1707 Mathematical Language and Symbols Language – is a complex system of words and symbols, either spoken or written, used by a particular community as a...

GE1707 Mathematical Language and Symbols Language – is a complex system of words and symbols, either spoken or written, used by a particular community as a means of communication. Language of Mathematics Every language has its vocabulary (the words) and its rules for combing these words into complete thoughts (the sentences). Math is comprised of primarily two things: numbers and symbols. Here are the most common mathematical symbols: Mathematical Symbols Symbol Meaning Example + add 3 + 7 = 10 − subtract 5−2 = 3 × multiply 4 × 3 = 12 ÷ divide 20 ÷ 5 = 4 / divide 20/5 = 4 () grouping symbols 2(𝑎𝑎 − 3) [] grouping symbols 2[ 𝑎𝑎 − 3(𝑏𝑏 + 𝑐𝑐) ] {} set symbols {1,2,3} 𝜋𝜋 pi 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2 P ∞ infinity ∞ is endless = equals 1+1 = 2 ≈ approximately equal to 𝜋𝜋 ≈ 3.14 ≠ not equal to 𝜋𝜋 ≠ 2 ≥ greater than, greater than or equal to 5 > 1 square root ("radical") √4 = 2 ° degrees 20° ∴ therefore 𝑎𝑎 = 𝑏𝑏 ∴ 𝑏𝑏 = 𝑎𝑎 Characteristics of the language of mathematics 1. Precise – able to make very fine distinctions; 2. Concise – able to say things briefly; 3. Powerful – able to express complex thoughts with relative ease. The language of mathematics can be learned but requires the efforts needed to learn any foreign language. English Language to Mathematical Language In English, nouns are used to name things we want to talk about (like people, places, and things); whereas sentences are used to state complete thoughts. A typical English sentence has at least one noun, and at least one verb. For example, consider the sentence: Carol loves mathematics. Here, ‘Carol’ and ‘mathematics’ are nouns; ‘loves’ is a verb. The mathematical analog of a ‘noun’ will be called an expression. Thus, an expression is a name given to a mathematical object of interest. Whereas in English we need to talk about people, places, and things, we’ll see that mathematics has many different ‘objects of interest’. 02 Handout 1 *Property of STI Page 1 of 4  GE1707 The mathematical analog of a ‘sentence’ will also be called a sentence. A mathematical sentence, just as an English sentence, must state a complete thought. The table below summarizes the analogy. ENGLISH MATHEMATICS name given to an object of NOUN (person, place, thing) EXPRESSION 1 interest: Examples: Carol, Manila, book Examples: 5 , 2 + 3 , 2 a complete thought: SENTENCE SENTENCE Examples: Examples: The capital of Philippines is Manila. 3+4=7 The capital of Philippines is Makati. 3+4=8 An expression is the mathematical analog of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. An expression does not state a complete thought; it does not make sense to ask if an expression is true or false. The most common expression types are numbers, sets, and functions. Numbers have lots of different names. For example, the expressions 10 5 2+3 (6-2)+1 1+1+1+1+1 2 All look different but all are names for the same number. A mathematical sentence is the analog of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. Sentences have verbs. In the mathematical sentence 3 + 4 = 7, the verb is “ = ”. A sentence can be always true, always false, or sometimes true/sometimes false. For example, the sentence 1 + 2 = 3 is true. The sentence 1 + 2 = 4 is false. The sentence 𝑥𝑥 = 2 is sometimes true/sometimes false: it is true when 𝑥𝑥 is 2, and false otherwise. The sentence 𝑥𝑥 + 3 = 3 + 𝑥𝑥 is always true, no matter what number is chosen for 𝑥𝑥. Sets, Functions, and Relations The word “is” has three (3) quite distinct meanings. The three (3) meanings are illustrated in the following three sentences. (1) 5 is the square root of 25 (2) 5 is less than 10 (3) 5 is a prime number The first sentence, “is” could be replaced by “equals”: it says that two objects, 5 and the square root of 25, are in fact one and the same object, just as it does in the English sentence “London is the capital of the United Kingdom.” In the second sentence, “is” plays a completely different role. The words “less than 10” form an adjectival phrase, specifying a property that numbers may or may not have, and “is” in this sentence is similar to “is” in the English sentence “grass is green.” As for the third sentence, the word “is” means “is an example of”, as it does in the English sentence “Mercury is a planet.” Another way, which doesn’t hide the word “is”, is to use the language of sets. 02 Handout 1 *Property of STI Page 2 of 4  GE1707 Sets A set is a collection of objects, and in mathematical discourse, these objects are mathematical ones such as numbers, points in space, or other sets. If we wish to rewrite sentence (3) symbolically, another way to do it is to define 𝑃𝑃 to be the collection, or set, of all prime numbers. Then (3) can be rewritten, “5 belongs to the set 𝑃𝑃”. This notion of belonging to a set is sufficiently basic to deserve its own symbol, and the symbol used is ∈. So a fully symbolic way of writing the sentence is 5 ∈ 𝑃𝑃. The members of a set are usually called its elements, and the symbol ∈ is usually read “is an element of”. So the “is” of the sentence (3) is more like ∈ than =. Although one cannot directly substitute the phrase “is an element of” for “is”, one can do so if one is prepared to modify the rest of the sentence a little. Functions Focusing on the phrase “the square root of” in a sentence (1). If we wish to think about such a phrase grammatically, then we should analyze what sort of role it plays in a sentence. The analysis is simple: in virtually any mathematical sentence where the phrase appears, it is followed by the name of a number. If the number is 𝑛𝑛 then this produces the slightly longer phrase, “the square root of 𝑛𝑛”, which is a noun phrase that again denotes a number and plays a similar grammatical role to one (at least when the number is used in its noun sense rather than its “adjective” sense). For instance, replacing “five” by “the square root of 25” in the sentence “five is less than seven” yields a new sentence, “The square root of 25 is less than seven”, that is still grammatically correct (and true). One of the most basic activities of mathematics is to take a mathematical object and transform it into another one, sometimes of the same kind and sometimes not. “The square root of” transforms numbers into numbers, as do “four plus”, “two times”, “the cosine of” and “the logarithm of”. A non-numerical example is “the center of gravity of”, which transforms geometrical shapes (provided they are not too exotic or complicated to have a center of gravity) into points - meaning that if S stands for a shape, then “the center of gravity of S” stands for a point. A function is, roughly speaking, a mathematical transformation of such a kind. Relations Let us now think about the grammar of the phrase “less than” in a sentence (2). As with “the square root of”, it demands to be followed by a mathematical object. Once we have done this, we obtain a phrase such as “less than 𝑛𝑛” which is importantly different from “the square root of 𝑛𝑛” because it behaves like an adjective rather than a noun, and refers to a property rather than an object. This is just how prepositions behave in English: look for example at the word “under” in the sentence “The cat is under the table.” At a slightly higher level of formality, mathematicians like to avoid too many parts of speech, as we have already seen for adjectives. So there is no symbol for “less than”: instead, it is combined with the previous word “is” to make the phrase “is less than”, which is denoted by the symbol < in a sentence, one should precede it by a noun and follow it by a noun. For the resulting grammatically correct sentence to make sense, the nouns should refer to numbers (or perhaps to more general objects that can be put in order). A mathematical “object” that behaves like this is called a relation, though it might be more accurate to call it a potential relationship. “Equals” and “is an element of” are two other examples of relations. Elementary logic A logical connective (also called logical operator) is a symbol or a word which is used to connect two or more sentences. Each logical connective can be expressed as a truth function. Logical connectives 1. Negation – is the opposite of a statement, usually employing the word not. The symbol to indicate negation is ~. For example: Original Statement Negation of statement Today is Monday Today is not Monday. It is raining. It is not raining. 02 Handout 1 *Property of STI Page 3 of 4  GE1707 2. Conjunction – A conjunction is a compound sentence formed by using the word and to join two simple sentences. The symbol for this is ⋀. When two simple sentences, 𝑝𝑝 and 𝑞𝑞, are joined in a conjunction statement, the conjunction is expressed symbolically as 𝑝𝑝 ⋀ 𝑞𝑞. Simple Sentences Compound Sentence: Conjunction 𝑝𝑝: Mariella eats fries. 𝑝𝑝 ∧ 𝑞𝑞: Mariella eats fries and Mae drinks soda. 𝑞𝑞: Mae drinks soda. 3. Disjunction – A disjunction is a compound sentence formed by using the word or to join two simple sentences. The symbol for this is ⋁. When two simple sentences, 𝑝𝑝 and 𝑞𝑞, are joined in a disjunction statement, the disjunction is expressed symbolically as 𝑝𝑝 ∨ 𝑞𝑞. Simple Sentences Compound Sentence: Disjunction 𝑝𝑝: The clock is slow. 𝑝𝑝 ∨ 𝑞𝑞: The clock is slow or the time is correct. 𝑞𝑞: The time is correct. 4. Implication – is a type of relationship between two statements or sentences. The statement “𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑞𝑞” means that if 𝑝𝑝 is true, the 𝑞𝑞 must also be true, and it is symbolized by a double-lined arrow pointing toward the right ⇒.The statement “𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑞𝑞” is also written “𝑖𝑖𝑖𝑖 𝑝𝑝 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑞𝑞” or sometimes “𝑞𝑞 𝑖𝑖𝑖𝑖 𝑝𝑝”. Statement 𝑝𝑝 is called the premise of the implication and 𝑞𝑞 is called the conclusion. Consider the following statement: 𝑝𝑝: Helen finishes her homework. 𝑞𝑞: Helen cleans her room. There are different ways to write the conditional 𝑖𝑖𝑖𝑖 𝑝𝑝 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑞𝑞. Notice that the premise 𝑝𝑝 is connected to the word 𝑖𝑖𝑖𝑖 in the sample shown: 𝑝𝑝 ⇒ 𝑞𝑞: If Helen finishes her homework, then she will clean her room. 𝑝𝑝 ⇒ 𝑞𝑞: Helen will clean her room, if she finishes her homework. References: Carol, B. (N.D.) Language of Mathematics. Retrieved from http://www.onemathematicalcat.org/pdf_files/LANG1.pdf The language and grammar of mathematics. Retrieved from https://www.dpmms.cam.ac.uk/~wtg10/grammar.pdf Carol, B. (N.D.) Expressions versus sentences. Retrieved from http://www.onemathematicalcat.org/pdf_files/LANG1.pdf Logical Connectives. (N.D.) Retrieved from https://www.catsyllabus.com/logical-reasoning/logical-connectives Mathematical statements and truth values. Retrieved from http://www.mathwarehouse.com/math-statements/logic-and-truth-values.php 02 Handout 1 *Property of STI Page 4 of 4