MMW - Midterm Reviewer - 1st Semester Psychology 1A PDF

MATHEMATICS IN THE MODERN WORLD regularity in the world or in a man-made design.” - LESSON 1: Patterns and Numbers in Nature and (Collins, 2018) World Patterns NATURE of MATHEMATICS...

MATHEMATICS IN THE MODERN WORLD regularity in the world or in a man-made design.” - LESSON 1: Patterns and Numbers in Nature and (Collins, 2018) World Patterns NATURE of MATHEMATICS Are regular, repeated, or recurring forms 1. Patterns and Relationships - To or designs - a series or sequence that understand their potential connections. repeats and can be observed in various 2. Mathematics, Science & Technology - aspects of life. Widely applicable across various fields, THROUGH PATTERNS… due to its universal nature and ability to Understand the world model complex relationships. Predict future events 3. Mathematical Inquiry: - Abstracting Discover new things real-world problems, manipulating Analyze situations logically, and interpreting the results. Organize information 4. Abstraction and Symbolic Influence outcomes Representation: - Identifying common Pattern exists when a set of numbers, colors, features between different things and shapes, or sounds are repeated over and over using symbols to represent shared again aspects. Example: 5. Manipulating Mathematical Statements: Teachers - to enhance students’ - combined and rearranged through learning and development. specific rules. PAGASA - tell or predict occurrences. 6. Application: - provide insights into Police Department - to solve a crime. real-world things - accuracy varies on Financial Markets - to see stocks’ rise how well underlying phenomena are and falls. represented. Symmetry WHAT DOES MATHEMATICS HAVE TO DO divided into two identical mirror halves. WITH THE NATURE? “Regularity - same thing always happens in the same circumstances. Pattern - discernible  Number Patterns and Sequences TYPES of SYMMETRY 1. Reflection Symmetry - a.k.a. Line/Bilateral Symmetry. (simplest form of symmetry) Objects can be divided into two identical halves by an imaginary line. Importance of Mathematics in Life 1. Restaurant Tipping 2. Netflix film viewing 3. Calculating Bills 2. Rotational Symmetry - rotated around a 4. Computing Test Scores central point that appears 2 or more 5. Doing Exercise times called Order and is still identical to 6. Handling Money its original position. 7. Making Countdowns 3. Point Symmetry - Every part has a 8. Baking and Cooking matching part that is equidistant from 9. Surfing Internet the central point but in the opposite Appreciating Mathematics As A Human direction. Endeavor 1. Agriculturists 2. Architects 3. Biologists 4. Chemists  5. Computer Programmers 6. Engineers (Chemical, Civil, Electrical, Industrial, Material) 7. Lawyers 8. Managers 9. Medical Doctors 10. Meteorologists 11. Politicians 12. Salespeople Fibonacci Sequence 13. Technicians Fibonacci sequence - series of numbers where each number is the sum of the two How can math be so universal? preceding numbers (called Fibonacci According to Annenberg Learner (2017) number). Sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, First, human beings didn't invent math 21, 34, and so on. concepts; we discovered them. Mathematics - discovered not an invention and universally applicable in various aspects of life, (personal finance to societal issues) Population Growth - follows an exponential Introduction to Fibonacci Sequence pattern. Number Sequence (Middle Ages) Population growth ○ Named after Leonardo Pisano Increase in the number of individuals Bigollo (Italian mathematician) within a population - currently occurring - discovered Fibonacci. 1.05% rate per year globally. Fibonacci: ○ short term for filius bonnaci (latin) ○ Meaning - “the son of bonacci”. Development of Fibonacci Sequence  1202: Leonardo Pisano Bigollo - most Fib (9) = (n – 1) + (n – 2) prominent work, the Liber Abaci (The Fib (9) = (9 – 1) + (9 – 2) book of Calculating). Fib (9) = (X8 = 21) + (X7 = 13) ○ He introduced his famous rabbit Fib (9) = 34 problem. The Golden Ratio’ “If a pair of rabbits is put into a walled (a.k.a) “golden section or golden enclosure (room) to breed, how many proportion” pairs of rabbits will be produced in a mathematical constant approximately year under specific breeding conditions. equal to 1.618 (value of the golden which begins to bear young two months ratio, which is the limit of the ratio of after its own birth? ” consecutive Fibonacci numbers), representing the ratio between two quantities. Represented - Greek letter phi, Φ is sometimes called the "divine proportion," because of its frequency in the natural world. Finding the Fibonacci number Obtained from the formula Xn = (Xn-1) + (Xn-2) by adding the two previous numbers in the sequence. The Rule The first term (n) are numbered from 0 onwards, and the Fibonacci number (Xn), see figure below: BINET’S FORMULA Example 4: Find Fib (9) Solution:  symbols (π, =, , ≥.), to represent mathematical ideas. Convention in the Mathematical Language Standard or agreement among mathematicians regarding specific facts, terms, or symbols. LESSON 2: Mathematical Language and Symbols Is Mathematics a Language? Yes, it provides a system for expressing and manipulating concepts. Prime numbers - Natural numbers greater than It uses a symbolic language to represent 1 that has only 1 and itself as its factors. and communicate mathematical ideas. [ 11, 13, 17, 19 ] Composite number - more than two factors. Language of Mathematics: CHARACTERISTICS Precise. NUMBER TYPES ○ Accurate and specific. 1. NATURAL NUMBERS - “N” Concise. Positive whole numbers starting ○ Brief and efficient. from 1. Powerful. 2. INTEGERS - “Z” Expresses complex ideas with ○ precision and conciseness. Whole numbers including MATHEMATICS SYMBOLS positive, negative, and zero. Uses symbols, numbers, instead of 3. RATIONAL NUMBERS - “Q” words. Symbols that we commonly use Numbers that can be expressed are: The 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, as a fraction. and 9); symbols for basic operations (+, 4. REAL NUMBERS - “R” −, ×, /); symbols that "stand in" for values All rational and irrational called variables like x, y and other letters numbers, including positive, written in lowercase; and special  negative, and zero, and Verb - the equal sign “=”, or inequality transcendental numbers. like < or >. 5. IMAGINARY NUMBERS - “I” Pronouns - variables (x or y, say, 5x – 7, Numbers that, when squared, xy2, and -3/x) give a negative result. The "unit" Adjective - a subscript (“n” in Xn). imaginary number is √(-1), and its And Sentence, say, 3x + 7 = 22, which is symbol is I, or sometimes j, where a sentence in mathematics. i2 = -1. The = sign and its variants LETTER CONVENTIONS Another: equal (=) sign (frequently used Start: a, b, c, - constant (fixed values) symbol) - does not mean anything on its own - From i, j, k, l, m, n we need a context. ○ positive integers (for counting) Variations on the equals sign are: End: x, y, z - variables (unknown) ≠ means ‘is not equal to’ ≈ means ‘is approximately equal to’ Use of uppercase and lowercase letters: ≥ means ‘is greater than or equal’ 1. Uppercase letter - name a set., ≤ means ‘is less than or equal to’ a. A = {1, 2, 3, 4}. Variables 2. Lowercase letter - represent an element. another form of mathematical symbol. a. B = {a, b, c} These are used when quantities take NOUNS, VERBS, SENTENCES different values. The most commonly Mathematics doesn't use the terms used variables are x, y, and z. "noun," "verb," or "pronoun," just Greek letters similarities with these grammatical Pi (π) - mathematical constant number concepts but, mathematics doesn't 3.14159.... require subject-verb agreement, unlike Greek letters (α, β, θ) - represent angles. English Sigma (Σ) - summation notation for Nouns - fixed things (numbers, or adding multiple numbers. expressions with numbers) : ○ Examples: 12, 3(5-2/3) and 102  The positioning of numbers and symbols statement that can be determined as in relation to each other also gives true or false. 23 + x = 76 meaning. Contains verbs. Examples: True, false, or sometimes true. 2 Superscript X means squared of X Equations show equality. Characters with slightly different shapes Subject, verb, and object. and sizes. Complete thought. Example: 25o is different from 250. ❖ The mathematical sentence 3 + 4 = 7, ○ These two have different the verb is ‘=’. meanings. The first is an angle ❖ The sentence 1 + 2 = 3 is true. The measured in degrees, the second sentence 1 + 2 = 4 is false. represents a temperature ❖ The sentence x = 2 is sometimes reading. true/sometimes false: it is true when x is 2, and false otherwise. EXPRESSIONS VERSUS SENTENCES ❖ The sentence x + 3 = 3 + x is (always) 1. Mathematical Expression true, no matter what number is chosen Incomplete thought - Numerical for x. phrase without subject or verb. Not true or false. LESSON 3: BASIC CONCEPTS: SETS Combination of numbers, SETS variables, and operators. Georg Cantor (1845-1918) introduced Common types: numbers, sets, the word “SET” as a formal mathematical term functions. in 1879. Represents a single value. Shows For Georg Cantor it simply means the value of something. “Collection of Elements” Example: 23 + x. Set - collection of objects, distinct and 2. Mathematical Sentence, also called well-defined. mathematical statement, or simply This unordered collection of distinct statement, or proposal, is a complete objects are called elements.  SETS - Capital letters. UNIT SET ELEMENTS - Small letters. SET NOTATION Empty Set/Null Set A set may be stated in two ways: 1. Set-Roster Notation SUBSETS 2. Set-Builder Notation Set-Roster Notation all the elements between braces are written. Example 1: Set of natural numbers less than 6 Solution: A = {1, 2, 3, 4, 5} Set-Builder Notation PROPER SUBSET Example: Set of names of Grade I teachers in the province of Laguna that starts with letter A. (Note: Since it’s impossible to write all the names the set builder notation is more appropriate to use). Distinction between elements and subsets Solution: B = {x ∣ x is a Grade I teacher whose Elements - Objects in a set. name starts with A}. The symbol “∣” is read as Equal sets - Same elements. such as. Empty/null set - No elements. TYPES OF SETS Subset - Elements of one set are also in FINITE - Definite number of elements. another set. INFINITE - Set whose endless elements.  If every element in Set A is also an LESSON 4: OPERATIONS OF SETS element in Set B, then Set A is a subset of Set B. Finite set - if there are “n” distinct CARTESIAN PRODUCT numbers of elements. Cartesian product - Critical ○ Where n∈N breakthroughs by Norbert Wiener (1894 ○ CARDINALITY of SETS = number of – 1964) and Felix Hausdoff (1868 – 1942). distinct elements that is denoted Ordered pair - Defined by Wiener and by |A| = n Hausdoff but Incomplete definition. Example: Kazimierz Kuratowsk - Defined the ○ E = {a,b,c,d} - E = {4} ordered pair completely in 1921. ○ F = {1,1,2,3,3,4,4,4} - F = {4} {{a}, {a,b}} Infinite Set ○ B = {x∈ Z+} Power Set (P(n)) ○ set of all subsets of the set Example: ORDERED PAIR ○ A = {1,2,3} Given elements a and b, the symbol ○ P(A) = { }, {1}, {2}, {3}, {1,2}, {1,3}, (a,b) denotes the ordered pair consisting of a {2,3}, {1,2,3} and b together with the specification that a is ○ { } = null set or “Ø” empty set the first element of the pair and b is the second ○ Formula: 2n = where “n” is the sum element. Two ordered pairs (a,b) and (c,d) are of all the elements in the sets equal if, and only if, a=c and b=d. (a,b) = (c,d) Example: means that a=c and b=d ○ P(A) = 2(3) = 2*2*2 = 8 EXAMPLE: Universal Set (U) If A = {7, 8} and B = {2, 4, 6} ○ Usually denoted by the capital find (a) A × B letter ‘U’, sometimes by Solution: (a) A × B = {(7, 2); (7, 4); (7, 6); ε(epsilon). A set that contains all (8, 2); (8, 4); (8, 6)} the elements of other sets including its own elements.  ○ U = {counting numbers} ○ U = Set of integers Example: ○ A = {1, 2, 3} ○ B = {3, 4, 5} ○ C = {5, 6, 7, 8, 9} ○ U = {1, 2, 3, 4, 5, 6, 7, 8, 9} Complement (Nc) ○ all the objects that do not belong Union (U) to the other set. ○ Elements that belong to set A or ○ The complement of B with set B respect to A is the set of all ○ Must written in ascending order elements that belong to A but Example: not in B. Example: ○ A - B = {x|x∈A and x∉B} ○ A - B = { } “The complement of B = {} with respect to A = {} is equal to the set of = {}” Intersection (∩) ○ Elements that belong to both the sets, A and B - COMMON ELEMENT ○ Sum of both sets. Example 1: ○ Disjoint Sets - sets that contain no ○ A = {a, b, c} common elements. ○ B = {b, c ,d, e} Example: ○ A - B = {a} ○ B - A = {d, e} Example 2:  ○ A = {x|x ≤ 4} (... -1, 0, 1, 2, 3, 4) ○ A⋃B = B⋃A ○ U = Z (INTEGERS) ○ A⋂B = B⋂A ○ U - A = {x|x > 4} = {5, 6, 7, 8, 9…} 2. Associative Properties Example 3: ○ both union and ○ A = {x|x ≤ 4} (... -1, 0, 1, 2, 3, 4) intersection fulfill the ○ U = {x|0 < x < 10} associative property of ○ U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9} sets. Symmetric Differences (⊕) WHEREAS: ○ objects that belong to A or B but ○ (A⋃B)⋃C = A⋃(B⋃C) not to their intersection. ○ (A⋂B)⋂C = A⋂(B⋂C) ○ If the two sets are A and B, then 3. Distributive Properties symmetric differences belong to ○ A⋃(B⋂C) = (A⋃B)⋂(A⋃C) both A or B but not all. ○ A⋂(B⋃C) = (A⋂B)⋃(A⋂C) Example 1: 4. Idempotent ○ A = {a, b, c, d} ○ A⋃A = A ○ B = {a, c, e, f, g} ○ A⋂A = A ○ A⊕B = {b, d, e, f, g} 5. Complement (De Morgan’s Laws Example 2: ○ A⋂A’ = ∅ ○ S = {-7, 12, 76, 315, 426, 900} ○ A⋃A’ = U ○ T = {34, -7, 89, 315, 900} ○ For any two finite sets A ○ S⊕T = {12, 34 ,76, 89, 426} and B; Algebraic Properties of Sets (i) A – (B ∩ C) = (A – B) U (A – C) 1. Commutative Properties (ii)A – (B U C) = (A – B) ∩ (A – C) ○ Union and intersection ○ De Morgan’s Laws can operation satisfy the also be written as: commutative property, (i)(A ∩ B)’ = A’ U B’ which means that p+q= (ii) (A U B)’ = A’ ∩ B’ q+p in algebraic terms. WHEREAS:  Addition Principle 4. One output per input. ○ Cardinality (sum) of both Sets. 5. Distinct concepts. ○ if two sets of items are distinct Input (x-value) - domain. from one another (there is no Output (y-value) - range. overlapping), then the sum of the Functions - one output per input. union of the sets is obtained by Relations - may have multiple adding the sum of each set outputs. together. Determine by input-output Example 1: mapping. ○ A = {1, 2, 3} Function: ○ B = {4, 5 ,6} ○ AUB = |A|+|B|-|A∩B| ○ AUB = 3+3 - 1 ○ AUB = 6 - 1 ○ AUB = 5 Not a Function: LESSON 5: Basic Concepts: Relations and Functions Relation - a set of inputs and outputs that are related in some way. Relation exists between sets A and B if Function - a relation with one output for there is a connection or association each input. between their elements. Functions and Relations: 1. Relationships with meaning. Solution: 2. All functions are relations. 3. Not all relations are functions.  Using the notation x y which To determine explicitly the composition represent the sentence “x is not related to y”, of R, examine each ordered pair in A x B to see then: whether its elements satisfy the defining a) 1 1 because 1 is not less than 1 condition for R. b) 2 1 because 2 is not less than 1 c) 2 2 because 2 is not less than 2 Domain and Codomain: Relation - Subset of A x B. Related elements - (x, y) in R if x is related to y. Domain - Set A; Input values for a function. Co-domain - Set B; Possible output values. Range - Output values of a function. Solution: Is 1R3? Yes, 1 R 3 because (1, 3) ∈ R. Is 2R3? Relation: No, 2 3 because (2, 3) ∉ R. Is 2R2? Yes, 2 R 2 because (2, 2) ∈ R. Solution: domain of R is (1, 2) co-domain is (1, 2, 3). Circle Relation: Solution: A. A x B = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}.  Functions: Properties (1) and (2) can be stated less formally as follows: Solution: A relation F from A to B is a function if B. The domain and the co-domain of C are and only if: both R, the set of all real numbers. Every element of A is the first element of C. See the figure. an ordered pair of F. No two distinct ordered pairs in F have the same first element. Arrow Diagrams of Relation  Binary Operation: Example: Graph of Functions: Using Vertical Line Test, that is, a set of points in the plane is the graph of a function if and only if no vertical line intersects the graph in more than one point.  Simple statement - Single idea. Example of Simple Statement: ○ Daniel attends the opera concert. ○ Lyka performs in the opera concert. LESSON 6: Elementary Logic Compound statement - Multiple ideas Elementary Logic connected by connectives. Logic - Study of reasoning methods. Example of Compound Statement: Analysis of reasoning methods - Logic's Daniel attends the opera concert focus. or Lyka performs in the opera Form rather than content - Logic is concert. interested in form. Daniel attends the opera concert Mathematical logic - Symbolic or formal and Lyka performs in the opera logic. concert. Informal logic - Another branch of logic. Connectives - is used to combine two or more BRANCHES of LOGIC: propositions. Mathematical logic - Reasoning in mathematics. Informal logic - Reasoning outside formal settings. Example: All men are mortal. Luke is a man. Hence, Luke is mortal. All dogs like fish. Cyber is a dog. Negation Therefore, Cyber likes fish. Example: Proposition and Connectives Statement: Lyka performs in the opera. Proposition - True or false sentence. Negation: Lyka will not perform in the Declarative sentence - Another name opera. for proposition. Example of Preposition: concert or Gwen will not perform in the opera concert. ○ p^(q v ¬s) Quantifiers Express how many “objects” satisfy a Gwen performs in the opera concert given property or idea. and Kim attends the opera concert. Words like "all," "there exists," and "none." ^ ○ sq Quantified statement - statement with at Lyka performs in the opera concert or least one quantifier. Gwen performs in the opera concert. Variable - Represents an unspecified ○ rvs object. If Kim attends the opera concert Two types of quantifier - Existential then Gwen will perform in the opera quantifier and Universal Quantifier concert ○ Existential Quantifier - used to ○ q→s emphasize the existence of Lyka will perform in the opera concert if something. and only if Gwen will perform in the Words - "There exists," "at least one," "for opera concert. some." ○ r↔s Denying existence - "None" or "no." Daniel will not attend the opera concert. ○ ~p Either Daniel and Kim attend the opera concert or Lyka performs in the opera ○ Universal Quantifier - used to concert. T stress out every element that ○ (p^q) v r satisfies the condition. Daniel attends the opera concert Words - “all”, “for every” are used. and either Kim attends the opera Example: Charlie is telling the truth and Daniel is telling the truth. ○ 𝑐^𝑑 Bambi is lying then Charlie is telling the Negation of Statement truth. ○ b→c Alyssa is not lying and Daniel is telling the truth ○ ~𝑎^d The Truth Table And The Truth Value Truth table - Shows truth values of statements. Truth value - True or false. Example: Compound statement - Depends on Quantified Statement - “All freshmen simple statements and connectives. students are graduates of the K – 12 curriculum. Negation Statement - “Some freshmen students are not graduate of the K – 12 curriculum” Example: Convert the following mathematical symbols to English sentence based on the following simple propositions  T T F F T F T F T T T T T F F T T F F T T F F F F T F F F T F F T T T T F F F T T F [p^(q→r)]→(q→r) p q r q→r p^q p^(q→r) [p^(q→r)]→(q→r) T T T T T T T T T F F T F T T F T T F T T T F F T F T T Example: F T T T F F T p→¬(p^q) F T F F F F T p q p^q ¬(p^q) p→¬(p^q) F F T T F F T T T T F F F F F T F F T T F F T T F T F T T F F F T T (pV¬q)↔r p q r ¬q (pV¬q) (pV¬q)↔r T T T F T T