Mathematical Language and Symbols PDF

MATHEMATICAL LANGUAGE AND SYMBOLS This is used to construct and convey words as means of communication. It is used to deliver or send information and ideas. It can be an object, event, or even a person which represents something. The system used to communicate mathematical ideas. It consists...

MATHEMATICAL LANGUAGE AND SYMBOLS This is used to construct and convey words as means of communication. It is used to deliver or send information and ideas. It can be an object, event, or even a person which represents something. The system used to communicate mathematical ideas. It 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. Mathematical notation used for formulas has its own grammar and shared by mathematicians anywhere in the globe. USED TO WRITE: CHARACTERISTICS: Numbers Precise Sets Concise Functions Powerful Perform Operations The Ten Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Operations: +, -, /, * Variables: a, b, c, x and y Sets: ∪, ∩, ⊂, ⊃ Logic Symbols: ~, ∧, ∨, →, ↔ Special Symbols: =, , ≤, ≥, π Set Notations: N, W, Z, Q, R\Q, R, C name given NOUN EXPRESSION to an object (Aiah, cat) (1, 2, 1+1) of interest complete SENTENCE SENTENCE thought (Aiah owns a cat.) (1 + 1 = 2) An expression (or mathematical expression) is a finite combination of symbols defined by specific rules that vary by context. Symbols can designate numbers, variables, operations, functions, brackets, punctuations, and groupings. It is a correct arrangement of mathematical symbols used to represent a mathematical object interest. It does not state a complete thought. 2 2x + 5 7/8 x^2 A sentence (or mathematical sentence) makes a statement about two expressions, either using numbers, variables, or a combination of both. It is a correct arrangement of mathematical symbols that states a complete thought. It can be determined whether it is true, false, sometimes true/sometimes false. 3+2=5 2x + 5 < 11 x² > 9 t+3=3+t  MATHEMATICAL STATE A COMPLETE OBJECT OF INTEREST THOUGHT NUMBER TRUE SET FALSE FUNCTION SOMETIMES TRUE/ SOMETIMES FALSE Mathematical Convention is a fact, name, notation, or usage which is generally agreed upon by mathematicians. Symbols Sets Order of Operations Logical Statements Variables Quantifiers Functions Group Factor Axiom Ring Tensor Conjecture Field Fractal Theorem Term Functor Lemma Corollary Conventions in Mathematical Language Conventions in Mathematics Mathematical Conventions Mathematical Language’s Conventions C. 5y + 2 Precise, Concise, and Powerful 69 2x² + 2 or 2 + 2x² False Constant Sometimes True, Sometimes False + and - OR Addition and Subtraction ↔ Set theory is the branch of mathematics that studies sets or the mathematical science of the infinite. Georg Cantor (1845-1918) A set is a collection of objects. J = {x|x is a positive number greater than 2} A = {S, C, I, E, N, C, E} H = {x|x is an integers, 3 < x < 9} Roster Method- descriptive phrases to describe the elements in a set. Ruler Method- the elements are enumerated and separated by a comma (,). Finite and Infinite Sets Universal Set Unit Set Cardinality Empty Set FINITE SETS EXAMPLES INFINITE SETS EXAMPLES K = {x|x are letters in the A = {2, 4, 6, 8, …} word MATH} R = {x|x is a set of the I = {1, 2, 4, 8, 16} number of atoms in the Y = {x|x are integers, -5 < x earth} < 4} A = {x|x is a set of real numbers} J = {cat} O = {3} R= {x|x is a letter between a and c} L = {x|x is a planet where humans can live} M = {x|x is an integer less than 4 but greater than 3} A = {x|x are vowels in the word RHYTHM} Y={}  U = {1, 2, 3, 4, 5, …, 1000} U = {x|x is a letter in the alphabet} Denoted by the symbol U  It is the number of elements in a set. H = {a, b, c, d, e, f, g} n(H) = 7 E = {x|x is a planet in the solar system} n(E) = 8 Y = {r, e, d} n(Y) = 3 Introduced by John Venn in his paper "On the Diagrammatic and Mechanical Representation of Propositions and Reasoning’s"  If A and B are sets, A is called subset of B, if and only if, every element of A is also an element of B. Denoted as: A ⊆ B ↔∀x, x ∈ A → x ∈ B A = {c , d, e} B = {a, b, c, d, e} U = {a, b, c, d, e, f, g} Then A ⊆ B, since all elements of A is in B  Let A and B be sets. A is a proper subset of B, if and only if, every element of A is in B but there is at least one element of B that is not in A. The symbol ⊄ denotes that it is not a proper subset. Denotation: A ⊂ B ⟺ ∀x, x ∈ A → x ∈ B. A = {c, d, e} B = {a, b, c, d, e} C = {e, a, c, b, d} U = {a, b, c, d, e, f, g} Then A ⊂ B, since all elements of A is in B. A = B ⟺ A ⊆ B ∧ B ⊆ A. A = {a, b, c, d, e} B = {a, b, d, e, c} U = {a, b, c, d, e, f, g} Then A ⊆ B and B ⊆ A, thus A = B  Power set is the collection/set of all possible subsets of a set. Power set is denoted as P G = {1, 2} P(G) = {{1}, {2}, {1, 2}, {∅}} O = {a, c, e} P(O) = {{a}, {c}, {e}, {a, c}, {a, e}, {c, e}, {a, c, e}, {∅}} THEOREM 1.2: A SET WITH NO ELEMENTS IS A SUBSET OF EVERY SET: IF ∅ IS A SET WITH NO ELEMENTS AND A IS ANY SET, THEN ∅ ⊆ A. THEOREM 1.3: FOR ALL SETS A AND B, IF A ⊆ B THEN P(A) ⊆ P(B). THEOREM 1.4: POWER SETS: FOR ALL INTEGERS N, IF A SET S HAS N ELEMENTS, THEN P(S) HAS 2N ELEMENTS. UNION- A∪B = {x | x ∈ A ∨ x ∈ B} INTERSECTION- A∩B = {x | x ∈ A ∧ x ∈ B} COMPLEMENT- A’ = {x ∈ ∪ | x ∉ A} DIFFERENCE- A ~ B = {x | x ∈ A ∧ x ∉ B} = A∩B’ SYMMETRIC DIFFERENCE- A ⊕ B = {x | x ∈ (A∪B) ∧ x ∉ (A∩B)} = (A∪B)∩(A∩B)’ or (A∪B)~(A∩B). DISJOINT SETS- A and B are disjoint ⟺ A∩B = ∅ Order Pairs (a, b) where a is the first component and b is the second component.  Cartesian product of given sets. Suppose A = {1, 2, 4} and B = {5, 9}. Find each set. A relation is a set of ordered pairs. x corresponds to y or that y depends on x and is represented as the ordered pair of (x,y). A relation from set A to set B is defined to be any subset of AxB. If R is a relation from A to B and (a, b) ∈ R, then we say that “a is related to b” and it is denoted as a R b.  Let A = {a, b, c, d} be the set of car brands, and B = {s, t, u, v} be the set of countries of the car manufacturer. Then AxB gives all possible pairings of the elements of A and B, let the relation R from A to B be given by R = {(a, s), (a, t), (a, u), (a, v), (b, s), (b, t), (b, u), (b, v), (c, s), (c, t), (c, u), (c, v), (d, s), (d, t), (d, u), (d, v)}. Function is a special kind of relation helps visualize relationships in terms of graphs and make it easier to interpret different behavior of variables. A function is a set of ordered pairs (x,y) such that no two ordered pairs have the same x-value but different y-values. A = {(1, 3), (2,4), (3,5), (4,6)} B = {(-2,7), (-1,3), (0,1), (1,5), (2,5) C = {(3,0), (3,2), (7,4), (9,1)} A Binary Operation is an operation or rule that combines two elements to produce another element. a b = G, for all a, b, c G. A group is a set of elements, with one operation, that satisfies the following properties: Closure Property- The set is closed with respect to the operation. Associative Property- The operation satisfies the associative property. Identity Property- There is an identity element. Inverse Property- Each element has an inverse. If any two elements are combined using the operation, the result must be an element of the set. If (a*b)*c = a*(b*c) where a, b,c are an element of all real numbers then the binary operation is associative. Addition and Multiplication of real numbers is associative. Subtraction and Division of real numbers is not associative. There exists an elements e in G, such that for all a that is an element of G, a*e = e*a = a For each a in a group, there exists an element 1/a in a group. Such that a*1/a = 1/a*a = e Determine whether the set of all non-negative integers under addition is a group. Step 1: Closure Property, choose any two positive integers, 8 + 4 = 12 and 5 + 10 = 15. Step 2: Associative Property, choose three positive integers 3 + (2 + 4) = 3 + 6 = 9 (3 + 2) + 4 = 5 + 4 + 9 Step 3: Identity Property, choose any positive integer 8 + 0 = 8; 9 + 0 = 9; 15 + 0 = 15 Step 4: Inverse Property, choose any positive integer 4 + (-4) = 0; 10 + (-10) = 0; 23 + (-23) = 0 A statement/proposition is any declarative sentence that is either true (T) or false (F), but not both. We refer to T or F as the truth value of the statement. EXAMPLES: Baguio is the capital of the Philippines. Mathematics is an easy course. The sun is shining. 2+2=5 Propositional variables are typically represented by lowercase letters such as p, q, r. A sentences that maybe either single proposition or compound proposition. Composed of two or more simpler propositions, called atomic propositions, connected by logical operators: “not”, “and”, “or”, “if-then”, “if and only if” and “exclusiveor”. Negation Conjunction Disjunction Conditional Biconditional Exclusive-or  If p is true, then ~p is false. p = It is raining. ~p = It is not raining. p = Math is an easy course. ~p = Math is not an easy course. p = The world is flat. ~p = The world is not flat. p = The bank will not go bankrupt. ~p = The bank will go bankrupt. If p is true and q is true then p ^ q; otherwise, p ^ q is false. Let: Therefore: p = It is raining. p ^ q = It is raining and the sun is q = The sun is shining. shining. Let: Therefore: p = I will pass mathematics. p ^ q = I will pass mathematics and I q = I will graduate. will graduate. Let: Therefore: p=1+1=3 p ^ q = 1 + 1 = 3 and 2 + 2 = 5 q=2+2=5  If p is true or q is true or if both true, then p ˅ q is true; otherwise, p ˅ q is false. Let: Therefore: p = I will attend the meeting p ˅ q = I will attend the meeting or I q = I will send a representative. will send a representative. Let: Therefore: p = It is raining. p ˅ q = It is raining or it is cloudy. q = It is cloudy. Let: Therefore: p = You are going to the park. p ˅ q = You are going to the park or q = You are going to watch movies. you are going to watch movies.  If p is true, then q must also be true. In other words, p implies q. Let: Therefore: p = You are going to the movies. p → q = If you are going to the movies q = You are watching a romantic then you are watching a romantic comedy comedy Let: Therefore: p = I had a million pesos. p → q = If I have a million pesos then I q = I buy a house would buy a house Let: Therefore: p = You study hard. p → q = If you study hard then you will q = You pass the exam. pass the exam.  If p and q are true or false, the p ↔ q is true; if p and q have opposite truth values, the p ↔ q is false. Let: Therefore: p = It is raining in Manila. p ↔ q = It is raining in Manila if and q = It is cloudy in Manila. only if it is cloudy in Manila. Let: Therefore: p = A person is an adult. p ↔ q = A person is an adult if and q = They are at least 18 years old only if they are at least 18 years old. Let: Therefore: p = A person has a degree. p ↔ q = A person has a degree if and q = They have completed a only if they have completed a bachelor’s program. bachelor’s program.  If p and q are true or false then p ⊕ q is false; if p and q have opposite truth values, then p ⊕ q is true. Let: Therefore: p = It is raining in Manila on Monday. p ⊕ q = It is raining in Manila either on q = It is raining in Manila on Tuesday. Monday or on Tuesday, but not both. Let: Therefore: p = A car has four wheels. p ⊕ q = A car has either four wheels q = A car has six wheels. or six wheels, but not both. Let: Therefore: p = A person has a cat. p ⊕ q = A person has either a cat or a q = A person has a dog dog, but not both. An expression is completely formal when it is context independent and precise. Is a statement whose truth depends on the value of one or more variable. Predicates become proposition once every variable is bound by assigning a universe of discourse. Example: “x is a positive integer” is a predicate whose truth depends on value of x. Predicate can also be denoted by function-like notation Example: P(x) = “x is a positive integer” If P is a predicate, then P(x) is either true or false, depending on the value of x. A propositional function is a sentence P(x) that uses one or more variables and makes a statement that can be true or false. Propositional functions are denoted as P(x), Q(x), R(x), and so on. The independent variable of propositional function must have a universe of discourse, which is a set from which the variable can take values. “If x is an odd number, then x is not a multiple of 2.” There exists an x such that x is odd number and 2x is even number. For all x, if x is a positive integer, then 2x + 1 is an odd number. The universe of discourse for the variable x is the set of positive real numbers for the proposition. “There exists an x such that x is odd number and 2x is even number.” Binding variable is used on the variable x, we can say that the occurrence of this variable is bound. A variable is said to be free, if an occurrence of a variable is not bound. The scope of a quantifier is the part of an assertion in which variables are bound by the quantifier. A variable is free if it is outside the scope of all quantifiers. Existential quantifiers are a logical symbol used in mathematical logic and formal logic to express that there exists at least one element in a given domain that satisfies a particular property. The statement “there exists an x such that P(x),” is symbolized by Ǝx P(x). Symbol Ǝ The statement “Ǝx P(x)”is true if there is at least one value of x for which P(x) is true. Universal quantifiers are symbols used in logic to express that a certain property or condition holds for all elements within a given domain. The statement “for all x, P(x),” is symbolized by ∀x P(x). Symbol ∀ The statement “∀x P(x)”is true if only if P(x) is true for every value of x.  QUALIFIER TRANSLATION There exists There is some For some For which EXISTENTIAL Ǝ For at least one Such that Satisfying For all For each UNIVERSAL ∀ For every For any Given any If the universe of discourse for P is P {p1 , p2 , …, pn }, then ∀x P(x) ⟺ P(p1 ) ∧ P(p2 ) ∧…∧ P(pn ) and Ǝx P(x) ⟺ P(p1 ) ∨ P(p2 ) ∨…∨ P(pn ) STATEMENT Is True when Is False when There is at least one x ∀x P(x) P(x) is true for every x for which P(x) is false There is at least one x Ǝx P(x) P(x) is false for every x. for which P(x) is true. Statement Negation All A are B Some A are not B No A are B Some A are B Some A are not B All A are B Some A are B No A are B