Mathematical Language and Symbols: Chapter 2

• Chapter 2 discusses mathematical language and symbols.

Topic Outline

• Characteristics of Mathematical Language
• Expression versus Sentences
• Conventions in the Mathematical Language
• Four Basic Concepts
• Elementary Logic
• Formality

Conventions in Mathematical Language

• Mathematics uses spoken and written natural languages to express mathematical concepts.
• Mathematical language serves as a tool for mathematical expression, exploration, reconstruction after exploration, and communication.
• Mathematical language should be precise and concise.
• Poor understanding of the language can be an obstacle.
• Digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Includes Mathematical symbols.

Mathematical Language

• Mathematical language is communication of mathematical ideas.
• It uses natural language with technical terms and grammatical conventions which are supplemented by symbolic notation for formulas.
• Mathematical notation has its own grammar shared by mathematicians worldwide.
• Mathematical language is precise, concise, and powerful.

Expressions Versus Sentences

• Expressions are combinations of symbols defined by context-dependent rules.
• Symbols in expressions represent numbers, variables, operations, functions, brackets, punctuations, and groupings which show order of operations.
• Expressions do not present a complete thought and cannot be assessed as true or false; Examples of expressions include numbers, sets, and functions.
• Sentences are statements about expressions, using numbers, variables, or combinations of both, and symbols or words like equals, greater than, or less than.
• Sentences state complete thoughts that can be determined as true, false, or sometimes true/sometimes false.

Conventions in Mathematical Language

• Mathematical Convention is a fact, name, notation, or usage agreed upon by mathematicans
• PEMDAS (Parenthesis, Exponent, Multiplication, Division, Addition and Subtraction) is a mathematical convention.
• Terms like group, ring, field, term, and factor have specific meanings within mathematics.
• Special terms exist like tensor, fractal, and function.
• Mathematical taxonomy includes axiom, conjecture, theorems, lemma, and corollaries.
• Formulas read left to right, Latin letters denote variables/parameters.
• Common mathematical expressions:
  • = (equal)
  • • (addition)
  • ÷ (division)
  • ∃ (there exists)
  • ↔ (if and only if)
  • < (less-than)
  • • (subtraction)
  • ∈ (element)
  • ∞ (infinity)
  • ≈ (approximately)

  • (greater-than)

  • × (multiplication)
  • ∀ (for all)
  • → (implies)
  • ∴ (therefore)

Four Basic Concepts

• Language of Sets
• Language of Functions
• Language of Relations
• Language of Binary Operations

Language of Sets

• Set theory studies sets and the mathematical science of the infinite. George Cantor (1845-1918) founded set theory.
• A set is a well-defined collection of objects
• Objects in a set can be elements or members
  • ∈ means element of a set
  • ∉ means not an element of a set
• Examples of Sets:
  • A= {x|x is a positive integer less than 10}
  • B = {x|x is a real number and x² - 1 = 0}
  • C = {x|x is a letter in the word dirt}
  • D = {x|x is an integer, 1 < x < 8}
  • E = {x|x is a set of vowel letters} or E = {a, e, i, o, u}
• Examples of Sets vs. Non-Sets:
  • The list of course offerings of Centro Escolar University = Set
  • The elected district councilors of Manila City. = Set
  • The collection of intelligent monkeys in Manila Zoo = Not a set
• Examples of listing elements of Sets:
  • A = {x|xis a letter in the word mathematics}; A = {m, a, t, h, e, i, c, s}
  • B = {x|x is a positive integer, 3 ≤ x ≤ 8}; B = {3, 4, 5, 6, 7, 8}
  • C = {x|x = 2n + 3, n is a positive integer.}; C = {5, 7, 9, 11, 13, ...}
• Methods of writing sets:
  • Roster Method: elements are enumerated and separated by a comma (tabulation method)
    • E = {a, e, i, o, u}
  • Rule Method: describes members or elements by descriptive phrase, written {x| P(x)} (set builder notation)
    • E = {x|x is a collection of vowel letters}
• Examples of Roster Form for Sets:
  • A= {x|x is the letter of the word discrete} = {d, i, s, c, r, e, t}
  • B = {x|3 < x < 8, x ∈ Z} = {4, 5, 6, 7}
  • C= {x|x is the set of zodiac signs} = {Aries, Cancer, Capricorn, Sagittarius, Libra, Leo, ...}
• Given sets in Roster Form they can be written using the Rule method.
  • D = {Narra, Mohagany, Molave, ...} = {x|x is the set of non-bearing trees}
  • E= {DOJ, DOH, DOST, DSWD, DENR, CHED, DepEd,...} = {x|x is the set of government agencies.}
  • F = {Botany, Biology, Chemistry, Physics, ...} = {x|x is the set of science subjects.}

Terms on Sets

• Finite and Infinite Sets.
• Unit Set.
• Empty Set.
• Universal Set.
• Cardinality.
• Finite Set: A set with limited or countable elements where the last element can be identified.
  • A = {x|x is a positive integer less than 10}
  • C = {d, i, r, t}
  • E = {a, e, i, o, u}
• Infinite Set: A set with unlimited or uncountable elements where the last element can't be specified.
  • F = {..., -2, -1, 0, 1, 2,...}
  • G = {x|x is a set of whole numbers}
  • H = {x|x is a set of molecules on earth}
• Unit Set: Also called a singleton, a set with only one element.
  • I = {x|x is a whole number greater than 1 but less than 3}
  • J = {w}
  • K = {rat}
• Empty Set: Also known as a null set, a set with no elements denoted by Ø or {}.
  • L = {x|x is an integer less than 2 but greater than 1}
  • M = {x|x is a number of panda bears in Manila Zoo}
  • N = {x|x is the set of positive integers less than zero}
• Universal Set: Denoted by the symbol U, consists of all sets under investigation within set theory.
  • U = {x|x is a positive integer, x² = 4}
  • U = {1, 2, 3,...,100}
  • U = {x|x is an animal in Manila Zoo}
• Cardinality: A set's cardinal number is the number of elements or members in the set which is denoted by n(A).
  • E = {a, e, i, o, u}; n(E) = 5
  • A = {x|x is a positive integer less than 10}; n(A) = 9
  • C = {d, i, r, tt}; n(C) = 4
• Theorem 1.1: The uniqueness of the empty set means there is only one set with no elements.

Venn Diagram

• A Venn Diagram is a pictorial presentation of relations and operations on set.
• They can be called set diagrams which displays possible logical relations between finite collections of sets.
• Constructed with overlapping circles to comprise a collection of simple closed curves drawn on a plane.
• The interior of the circle represents members/elements, with the exterior presenting non-members.
• Introduced by John Venn in "On the Diagrammatic and Mechanical Representation of Propositions and Reasoning's".

Kinds of Sets

• Subset.
• Proper Subset.
• Equal Set.
• Power Set.
• Subset: If A and B are sets, A is a subset of B only if every element of A is also an element of B; Symbolically represented as A ⊆ B ∀x, x ∈ A → x ∈ B.
• Proper Subset: A is a proper subset of B if every element of A is in B, but B has at least one element not in A; A ⊊ B ∀x, x ∈ A → x ∈ B.
  • The symbol ⊄ means that it is not a proper subset.
• Equal Sets: Set A equals Set B if every element of A is in B and every element of B is in A and is Symbolically represented as A = B A ⊆ B ∧ B ⊆ A.
• Power Set: Given S from universe U, the power set of S, (S) is the collection (or sets) of all S’s subsets.
• Theorem 1.2: A set with no elements is a subset of every Set. If Ø is a set with no elements and A is any set, then Ø ⊆ A.
• Theorem 1.3: For all sets A and B, if A ⊆ B then (A) ⊆ (B).
• Theorem 1.4: Power Sets: For all integers n, if a set S has n elements then (S) has 2^n elements.

Operations on Sets

• Union.
• Intersection.
• Complement.
• Difference.
• Symmetric Difference.
• Disjoint Sets.
• Ordered Pairs.
• Union: The union of A and B, A∪B is the set of all x elements in U such that x is in A or x is in B, represented by the symbol: A∪B = {x|x ∈ A ∨ x ∈ B}.
• Intersection: The intersection of A and B, A∩B, is the set of all elements x in U such that x is in A and x is in B, represented by the symbol: A∩B = {x|x ∈ A ʌ x ∈ B}.
• Complement: The complement of A (or absolute complement of A), A' is the set of all x elements in U such that x is not in A which is expressed as: A' = {x ∈ U | x ∉ A}.
• Difference: The difference of A and B (or relative complement of B in regards to A), A ~ B, is the set of all x elements in U such that x is in A and x is not in B that is expressed as: A ~ B = {x|x ∈ A ʌx ∉ B} = A∩B'.
• Symmetric Difference: Given two sets A and B, their symmetric difference as the set includes each element belonging to either A or B, but not to both and is expressed as: A ⊕ B = {x|x ∈ (A∪B) ʌ x∉ (A∩B)} = (A∪B) ∩(A∩B)' or (A∪B) ~ (A∩B).
• Disjoint Sets: Two sets with no elements in common and are non-intersecting: symbolically: A and B are disjoint ↔ A∩B = Ø.
• Order Pairs: In the ordered pair (a, b), a is called the first component and b is called the second; generally (a,b) ≠ (b,a).
• Cartesian Product: The Cartesian product of sets A and B, is written as AxB = {(a,b) | a ∈ A and b ∈ B}.

Language of Functions and Relation.

• A relation is comprised of ordered pairs.
• If x and y are elements of sets and related, x corresponds to y/y depends on x, and is represented as the ordered pair (x, y).
• A relation ranges from set A to set B defined as some subset of AxB.
• If R is a relation from A to B, and (a, b) ∈ R, "a is related to b" which is shown as a R b.
• Domain of Relation: dom R = {a ∈ A|(a, b) ∈ R for some b ∈ B}
• Image/Range of R: im R = {b ∈ B|(a, b) ∈ R for some a ∈ A}
• Functions is relation which helps to interpret the behavior of variables in terms of graphs.
• Function applications includes financial applications, economics, medicine, engineering, sciences natural disasters, calculating pH levels, measuring decibels and designing machineries.
• A function is a relation in which there is exactly one value for each ordered pair's first component.
• The set X is the function's domain.
• With each element x in X, corresponding element y in Y is the image of x and can be called the value of the function at x.
• The set of the domain elements' images is called the range of the function can map from one set to another.
• Algebraic structures investigates sets associated with single operations which satisfy defined axioms.
• An operation on a set produces generalized structures such as adding integers, or matrix multiplication of invertible 2x2 matrices.
• The algebraic structures known as group.
• Binary operation: G is a function when a binary operation is applied on G, each ordered G element pair is assigned. Symbolically, a * b = G, for all a, b, c ∈ G.
• Group: A set of elements with an operation satisfying i) closure, ii) associative property, iii) identity element, iv) inverse.
  • Closure Property: Two elements combined using an operator result in an element that is still in the same set which is expressed as a * b = c ∈ G, for all a, b, c ∈ G.
  • Associative Property: A number of terms can be re-grouped without issue which is expressed as: (a * b) * c = a * (b * c), for all a, b, c ∈ G.
  • Identity Property: There is an element e in G, such that ae = ea and there exits an Identity property.
  • Inverse Property: For each a in G exists an element a^-1 of G, such that a * a^-1 = a^-1 * a = e
• Check if a set of all non-negative integers under addition is a group through the four properties:
  • Take the Closure property, positive integers e.g. 8 + 4 = 12 and 5 + 10 = 15, confirm that results are always members of the set.
  • Take the Associative property, e.g. 3 + (2 + 4) = 3 + 6 = 9 and (3 + 2) + 4 = 5 + 4 = 9.
  • The Identity property, e.g. 8 + 0 = 8; 9 + 0 = 9; 15 + 0 = 15, confirms that results are equivalent.
  • Take the Inverse property, e.g. 4 + (-4) = 0; 10 + (-10) = 0; 23 + (-23) = 0 which is inverse to a^-1 = -a.
  • Therefore the set of all non integers under addition is a group because 4 required properties are satisfied

Formal Logic

• Formal logic is the science or study of argument and reasoning evaluation.
• Formal logic differentiates correct reasoning from poor reasoning.
• The methods of reasoning
• Mathematical logic studies logic and its application to mathematics like deductive formal proofs, systems, and expressive formal systems.
• Areas of study include Set Theory, Recursion Theory, Proof Theory, Model Theory. Aristotle (382-322 BC) is the Father of Logic.
• The study of logic started in the late 19th century, in the context of the development of axiomatic frameworks for analysis, geometry, and arithmetic.

Statements

• A statement (or proposition) is a declarative sentence that is either true or false, but not both. The truth value is the truth/falsity.
• Identify statement:
  • Manila is the capital of the Philippines is true.
  • What day is it? is not a statement.
  • Help me, please is not a statement.
  • He is handsome is not a statement..
• Examples of ambiguous statements:
  • Mathematics is fun.
  • Calculus is more interesting than Trigonometry.
  • It was hot in Manila.
  • Street vendors are poor.

Propositional Variable

• A propositional variable is a variable used to represent a formal statement in propositional logic, for example, p, q, and r.
• Logical connectives combine simple statements to compound statements:
• Compound statements are two simple statements and connected by logical connectives, such as "or", "not", "if then", "and", "exclusive-or". "if and only if".
• A non compound statement is named simple/atomic statements.
• The conjunction of statements p/q is "p and q" and pq, is used when "^" is the symbol; If p is true and q is true, then p^q is true.
• Examples of true or false for Conjunction:
  • 2 + 6 = 9 and man is a mammal. = False
  • Manny Pacquiao is a boxing champion and Gloria Macapagal Arroyo is the first female Philippine President. = False
  • Ferdinand Marcos is the only three-term Philippine President and Joseph Estrada is the only Philippine President who resigned. = True
• The disjunction of the statement p and q is "p or q" and is expressed as pv q.
• Property: true if p is true/q is true/both are true otherwise pvq is false.
• For each of the following disjunction, evaluate its truth value.
  • 2 + 6 = 9 or Manny Pacquiao is a boxing champion = True
  • Joseph Ejercito is the only Philippine President who resigns or Gloria Macapagal Arroyo is the first female Philippine President = True
  • Ferdinand Marcos is the only three-term Philippine President or man is a mammal = True
• Negation of p, denoted by ~p is read "not p" and is shown as:
  • p, T ~p F
  • p, F ~p T
• Examples of negation:
  • Statement: 3 + 5 = 8 and Negation as 3 + 5 ≠ 8
  • Statement: Sofia is a girl and Negation as Sofia is a boy
  • Statement: Achaiah is not here and Negation as Achaiah is here

Conditional

• In "if p then q", is the conditional of p and q with pq.
• Term "p" is hypothesis, antecedent/premise and it is "if" statement.
• Term "q" is the conclusion, consequent/consequence and is "then" statement.
  • p T q T, p→q T
  • p T q F, p→q F
  • p F q T, p→q T
  • p F q F, p→q T
• In a conditional "if vinegar is sweet, then sugar is sour"
  • "Vinegar is sweet" is the antecedent.
  • "Sugar is sour" is the consequent.
• If vinegar is sweet, then sugar is sour is true
• 2 + 5 = 7 is and if is only if + 6= 1 is false.
• 14-8 = 4 means 6+ = 3 = 2. is true
• In a Biconditional: a statement is formed p ↔ q, a statement if p and only if q and is described as
  • p T q T, p ↔ q T
  • p T q F, p ↔ q F
  • p F q T, p ↔ q F
  • p F q F, p ↔ q T
• Exclusive-Or ( ⊕ ): formed when either p or q is true but if and only if, then is described as:
  • p T q T, p ⊕ q F
  • p T q F, p ⊕ q T
  • p F q T, p ⊕ q T
  • p F q F, p ⊕ q F
• "Sofia will take her lunch in Batangas or she will have it in Singapore."
  • If Sofia won't have lunch in both Batangas and Singapore: false.
  • If Sofia will only have Lunch in one location Batangas or Singapore, then it’s logically sound: as true
  • If she will have Lunch, but if it is elsewhere, like in Singapore: it is false

Predicate

• A predicate: A predicate (or open statements) is a statement whose truth depends on the value of one or more variables.
• Functions: functions become propositions once every variable is bound by assigning a universe of discourse and predicates are define in terms of its.
• E.g., x is an even number means the truth depends on x which depends on what that value is.
• P(x) x is an even number e.g. P(2) then the statement is true. If we substitute it with P(3) the state is false

Propositional Functions

• Sentence P(x); where statement happens, with value from independent variable that possess universe of discourse
• The function may be denoted as P(x), or there could be more Q(x), R(x), etc E.g, if x odd #, then x = 2 is in Logical (P(x)-> Q(x) specific, determine validity/logic, and be specific

Quantifiers

• For Existential case it is the sentence is true to P: exists x such that x is odd w 2x exists is even #
• The Universe - variable (binding/free) for statement by quant/assgmets
• And can bind Quant (exist/universal) , so free
• The scope is assertion, quantify ∃: exists x and symbol and state P
• When it's is it and if + its all case

Universal

• Is in this format “each x, be true
• All for each , its True only with x

Other

• In statement and the negation
• All A B = Some A Not B and etc..