MATM111-reviewer-prelim.pdf

lOMoARcPSD|46746737 MATM111-Prelims - aaaaa Mathematics in the Modern World (Our Lady of Fatima University) Scan to open on Studocu Studocu is not sponsored or endorsed by any college or univers...

lOMoARcPSD|46746737 MATM111-Prelims - aaaaa Mathematics in the Modern World (Our Lady of Fatima University) Scan to open on Studocu Studocu is not sponsored or endorsed by any college or university Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 aaaaaaaaaaaNATURE OF MATHEMATICSaaaaaaaaaa - When a figure moves from one place to another on a coordinate plane, a transformation occurs (the original figure 1. Patterns and Numbers in Nature and in the World changed position). 2. Fibonacci Sequence It is common practice to name original shapes using capital 3. Golden Ratio letters. (insert drawing) aaaaaaaaaaPATTERNS AND NUMBERS IN NATURE aaaaaaaaaaa It is common practice to name transformed shapes using the aaaaaaaaaasasaassaaaaAND IN THE WORLD aaaaaaaaaaaaaaaaaaaa same letters but with a “prime symbol”. (insert drawing) Patterns in nature are visible regularities of form found in the Four Basic Types of Transformation: natural world and can also be seen in the universe. Translation (SLIDE) Nature patterns which are not just to be admired, are vital clues - It slides each point (or vertex) of a figure at the same to the rules that govern natural processes. distance and in the same direction. - Nothing changes about the figure except for its position on Mathematical patterns can take many different forms, including the coordinate plane. numerical patterns, visual patterns, logic patterns, word patterns, and - The image is the same size, the same shape and it’s more. The number pattern is the most frequently used one. pointing in the same direction as the original. - Examples: Animals that hop or jump: kangaroo, rabbit, Image Pattern frog, grasshopper, etc. - Regular - (insert drawing) - Repeated Reflection (FLIP) - Recurring forms or designs - It flips a figure over a line called a line of reflection. 1. Tessellation - A figure and its reflection have the same shape and size, - A repeating pattern using regular polygons (figures with sides). but the figures face in opposite directions–like a mirror - No gaps or overlaps. image. - Each vertex is the same. - Examples: Light reflection - An illusion that tricks your mind and somehow moves based on - (insert drawing) the orientation of the images. Rotation (TURN) M. C. Escher - a famous artist who used tessellations in his - A transformation that turns a figure about a fixed point is works. called the center of rotation. Tessellation in Nature - examples are beehives, pineapple, and - An object and its rotation are the same shape and size, but turtle shells (carapace and plastron) the figures are turned in different directions. 2. Transformations - It can rotate clockwise or counterclockwise. - In geometry, a transformation also means to change. - (insert drawing) Dilation (SIZE CHANGE) Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 - A transformation that produces an image that is the same shape as the original, but is a different size. Sequence - A good real-life illustration of dilation would be our pupils - A particular order in which related events (or a sequence of and how they react to sunlight. events), movements, or things follow each other. It can be both Our pupils reduce in size (or get smaller) when we finite or infinite. are in the sunlight. In mathematics, it is a chain of numbers (or other objects) that usually Our pupils enlarge in size (or get bigger) when it’s follow a particular pattern. dark. - In other words, a sequence is a set of things (usually numbers) - (insert drawing) that are in order. 3. Fractal - The individual elements in a sequence are called terms. - Arrangement of things. - (insert drawing) ung may 3, 5, 7, 9 ganon - A never-ending pattern, infinitely complex patterns that are - “...”, triple dot, or three dots, is called ‘Ellipsis’ –means that it self-similar across different scales. is continuous but provided that you follow the pattern. - They are created by repeating a simple process over and over 1. Fibonacci Sequence in an ongoing feedback loop. Leonardo Fibonacci - The scale/size becomes bigger and bigger but the shape stays - He introduced the Fibonacci Sequence. the same (for example, cactus). - His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy. Logic Pattern - “Fibonacci” was his nickname, which roughly means “Son - Finding logical connections to form generalizations. of Bonacci”. 1. Non-verbal Reasoning Tests - As well as being famous for the Fibonacci Sequence, he - Are often used to assess the ability to solve problems and to helped spread Hindu-Arabic Numerals (like our present think analytically. numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place - It does not require prior knowledge. of Roman Numerals (I, II, III, IV, V, etc.). That has saved - Examples are: diagrammatic reasoning, logical reasoning, and us all a lot of trouble. inductive reasoning. - The mathematical properties of the nautilus shell represent (insert drawing) pervasive regularities in nature. - Fibonacci number: is an integer in the infinite sequence 1, 1, Number Pattern 2, 3, 5, 8, 13, … of which the first two terms are 1 and 1 each - Arithmetic Pattern/Sequence succeeding… - Geometric Pattern/Sequence - The Fibonacci Sequence was discovered after an investigation - Fibonacci Pattern/Sequence on the reproduction of rabbits. Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 - It can be written as a “Rule”. First, the terms are numbered Formula #2: from 0 onwards like this: 1+ 5 𝑛 ( 2 ) 𝑥𝑛 = n= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 … 5 xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 … - So term number 6 is called x6 (which equals 8). - Simple Rule: Add the last two terms to get the next. The Golden Ratio - It can be expressed as the ratio between two numbers. 𝑎 𝑎+𝑏 𝑏 = 𝑎 - (insert new formula ung may p) - p = 1.618 2 ways to complete the numbers of the Fibonacci Sequence: Formula #1: NOTE: The final answer should be constant (no decimal / round off the decimal). 2. Arithmetic Sequence - The nth term of an arithmetic sequence is given by: 𝑡𝑛 = 𝑎 + (𝑛 − 1)𝑑 Example: Find the 14th term of the arithmetic sequence 4, 7, 10, 13, … t14 = 4 + (14 - 1) 3 4 + 39 = 43 t14 = 4 + (13) 3 The 14th term in this sequence is the 43! Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 LESSON 1 (cont.) Difficulties in the Math Language: 1. Different meaning or use of words in Math and English. Language of Mathematics “and” is equivalent to plus. Language “is” may have a different meaning. - A system of conventional spoken, manual (signed), or written 2. The different uses of numbers: cardinal, ordinal, or nominal. symbols by means of which human beings, as members of a social “Cardinal numbers” are used for counting. group and participants in its culture, express themselves. “Ordinal numbers” are used for telling positions. “Nominal numbers” are used only as a name to identify Symbols commonly used in Mathematics are: something. Examples: Translation - English to Math Language 1. Six less than twice a number is forty-five. 2. A number minus seven yields ten. 3. A total of six and some number. 4. Twelve added to a number. 5. Eight times a number is forty-eight. 6. The product of fourteen and a number. ENGLISH Language VS MATHEMATICS Language 7. Twice a number minus eight. 8. The quotient of a number and seven is two. English Mathematics 9. Three-fourths of a number. Symbols English alphabet and English alphabet, 10. The product of a number and ten is eighty. punctuations numerals, Greek 11. Eight less than a number is five. letters, grouping 12. How many times does five go into twenty? symbols, special symbols Remember these keywords! Name Noun Expressions Complete Thought Sentence Equation Action Verbs Operations and other actions (simplify, rationalize) What is in a sentence Verbs Equality, inequality, membership in a set Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 Cardinality - tells how many things are in a set. When counting a set of objects, the last word in the counting sequence names the quantity for that set. Example Cardinality A = {5} |A| = 1 B = {7, 2} |B| = 2 End of Lesson 1 C = {1, 3, 4} |C| = 3 D = {9, 1, 5, 8} |D| = 4 E = {5, 5, 5, 5, 5} |E| = 1 LESSON 2 Equivalent Sets Set - Two finite sets A and B are equivalent if their cardinal number is - In mathematics, a set is a well-defined collection of distinct the same. objects, considered as an object in its own right. - n(A) = n(B) - Sets are one of the most fundamental concepts in mathematics. - A = {4, 5, 6} and B = {a, b, c} are equivalent. It is usually represented by capital letters. The objects of a set are separated by commas. Equal Sets The objects that belong in a set are the elements, or members of - Two sets A and B are said to be equal if every element of A is a the set. member of B, and every element of B is a member of A. It can be represented by listing its element between braces. - A = {4, 5, 6} and C = {6, 5, 4} are equal. A set is said to be well-defined if the elements in a set are - A=C specifically listed. Examples: 1. A = {a, e, i, o, u} 2. B = {set of plane figures} 3. C = {Ca, Au, Ag} One-to-One Correspondence - students can connect one number with one object and then count them with understanding. Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 4. Universal Set (U) - all members Solution: Use a Venn Diagram 1. Intersection (𝐴 ∩ 𝐵) - members of both set A and set B 5. Subset (𝐴 ⊂ 𝐵) - all members of set A are in set B 2. Union (𝐴 ∪ 𝐵) - members of set A or set B or both 6. Empty Set - (𝑛(ϕ) = 0) 3. Complementary (A’) - members not in the set Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 Functions and Relations Relation - any set of ordered pairs Function - a relation between a set of inputs having one output each. Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 End of Lesson 2 Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 LESSON 3 1. Negation of p Elementary Logic - Let p be a proposition. Logic - The statement “It is not the case that p” is also a proposition, called - It serves as a set of rules that govern the structure and presentation the “negation of p” or ¬p (read “not p”). of mathematical proofs. - It allows us to determine the validity of arguments in and out of mathematics. Table 1. The Truth Table for the Negation of a Proposition Proposition p ¬p - A statement that is either true or false, but not both. (it cannot be a question) T F - They can be expressed in symbols P, Q, R, or p, q, r. F T Simple - means single idea statement Compound - conveys two or more ideas Examples: p = The sky is blue. Examples: ¬p = It is not the case that the sky is blue. Atlanta was the site of the 1996 Summer Olympic games. (TRUE) ¬p = The sky is not blue. 1 + 1 = 2 (TRUE) 3 + 1 = 5 (FALSE) 2. Conjunction of p and q What will my MATM 111 grade be? (NOT A PROPOSITION) - Let p and q be propositions. - The proposition “p and q,” denoted by p∧q is true when both p and Logical Connectives q are true and is false otherwise. Statements Connectives Symbolic Types of Form Statements Table 2. The Truth Table for the Conjunction of two propositions. Not P Not ∼P Negation p q p∧q P and Q And P∧Q Conjunction T T T T F F P or Q Or P∨Q Disjunction F T F F F F If P, then Q If…Then P→Q Conditional P if and only if If and only if P↔Q Bi-conditional q Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 Example: p: 2 is an even integer q: 3 is an odd integer. The conjunction of these propositions is “2 is an even integer and 3 is an odd integer.” p: 2 is an odd integer q: 3 is an odd integer. The conjunction of these propositions is FALSE. 3. Disjunction of p and q - Let p and q be propositions. - The proposition “p or q,” denoted by p∨q, is the proposition that is false when p and q are both false and true otherwise. Table 3. The Truth Table for the Disjunction of two propositions. p q p∧q T T T T F T F T T F F F Example: p: students who have taken Algebra can take Discrete Mathematics. Example#2: q: students who have taken computer science can take Discrete p: 12 is prime = FALSE Mathematics. q: 17 is prime = TRUE The disjunction of these propositions is “Students who have r: 19 is composite = FALSE taken Algebra or Computer Science, but not both can take Discrete a. p ∨ q - 12 is prime or 17 is prime = TRUE Mathematics.” b. p ∨ r - 12 is prime or 19 is composite = FALSE c. q ∨ r - 17 is prime or 19 is composite = TRUE Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 4. Implication / Conditional p→q 4. If sugar is placed in water, then it will dissolve. - Let p and q be propositions. (Causal Implication) - The implication p→q is the proposition that is false when p is true 5. If x = 0 → y = 10, x ≠ 0 → y ≠ 10. and q is false, and true otherwise. (Material Implication) - In this implication p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence). Converse, Inverse, and Contrapositive Conditional p→q If p, then q Table 5. The Truth Table for the Implication of.p→q. Statement p q p→q Converse If q, then p q→p T T T Inverse ∼p → ∼q If not p, then not q T F F F T T Contrapositive ∼q → ∼p If not q, then not p F F T Now we can define the converse, the contra-positive and the inverse of Implications: a conditional statement. We start with the conditional statement “If P If p, then q then Q.” In symbol: (P → Q) p implies q The converse of the above conditional statement is “If Q then P.” If p, q In symbol: (Q → P) p only if q The contra-positive of the above conditional statement is “If not Q p is sufficient for q then not P.” In symbol: (∼Q → ∼P) q if p The inverse of the above conditional statement is “If not P then not q whenever p Q.” In symbol: (∼P → ∼Q) q is necessary for p Example: Examples: 1. Conditional Statement - “If it rained last night (P), then the 1. If all men are mortal then Mario is mortal. sidewalk is wet (Q).” (Logical Implication) Converse (Q → P) of the conditional statement is “If the sidewalk 2. If this polygon is a quadrilateral then it has four sides. is wet, then it rained last night.” (Definitional Implication) Contra-positive (∼Q → ∼P) of the conditional statement is “If the 3. If today is Thursday then tomorrow is Friday. sidewalk is not wet, then it did not rain last night.” (Sequential Implication) Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 Inverse (∼P → ∼Q) of the conditional statement is “If it did not rain last night, then the sidewalk is not wet.” 2. Conditional Statement - Hypothesis, p: it is raining; Conclusion, q: grass is wet. Converse statement (Q → P): “If the grass is wet, then it is raining.” Contra-positive (∼Q → ∼P): “If the grass is NOT wet, then it is NOT raining.” Inverse (∼P → ∼Q): “If it is NOT raining, then the grass is NOT wet.” End of Lesson 3 Inductive Reasoning - Is the process of reaching a general conclusion by examining specific examples. - Observations → Analysis → Theory LESSON 4 I break out when I eat peanuts. This is a symptom of being Problem Solving and Reasoning allergic. I am allergic to peanuts. 3, 6, 9, 12, 15, ? - Each successive number is 3 larger than the Inductive Reasoning preceding number. Thus, we predict that the next number in the - From a number of observations, a general conclusion is drawn. list is 3 larger than 15, which is 18. Members of a species are not all the same. - A statement is a true statement provided that is true in all cases. Individuals compete for resources. If you can find one case for which a statement is not true, Species are generally adapted to their environment. called a counterexample, then the statement is a false statement. Deductive Reasoning Every number that is multiple of 10 is divisible by 4. 100, 110, - From a general premise, specific results are predicted. 120, 130, 140, 150, 160, 170, 180, 190, 200. 100/4 = 25, 120/4 Individuals most adapted to their environment are more likely to = 30, 110/4 = 27.5. survive and pass their traits on to the next generation. All odd numbers under 10 are prime numbers. 1, 3, 5, 7, 9. 9 is not a prime number, therefore, it is a counterexample. Deductive Reasoning - Is the process of reaching a conclusion by applying general; assumptions, procedures, or principles. - Idea → Observations → Theory Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 All men are mortal. Jason is a man. Jason is mortal. All arguments are either valid or invalid. An argument is valid if and only if it is not possible that all of its Key Terms: premises are true and its conclusion is false. 1. Argument - is the reason or reasons offered for or against An argument is invalid if and only if it is possible that all of its something. premises are true and its conclusion is false. 2. Premises - minor or major propositions 3. Syllogism - an argument composed Deductively invalid arguments: generalization (induction), 4. Conclusion abduction. Examples: Examples: All rabbits in the hat are white. If a number is divisible by 2, then it must be even. 12 is divided These rabbits are from the hat. by 2. Therefore, 12 is an even number. → These rabbits are white. (Deduction, valid) These rabbits are from the hat. Inductive Reasoning Deductive Reasoning specific / observations generalization These rabbits are white. → (Probably) All rabbits in the hat are white. (Generalization, Every quiz has been easy. The catalog states that all Induction, invalid) Therefore, the test will be easy. entering freshmen must take a mathematics placement test. All rabbits in the hat are white. You are an entering freshman. These rabbits are white. Therefore, you will have → (Probably because) These rabbits are from the hat. (Abduction, to take a mathematics invalid) placement test. The teacher used PowerPoint in 90% of humans are right handed. Valid Invalid the last few classes. Therefore, Jose is human, therefore Joe is the teacher will use PowerPoint right handed. All beagles are dogs. All beagles are dogs. tomorrow. Snoopy is a beagle. Lassie is a dog. So, Snoopy is a dog. So, Lassie is a beagle. Every fall… Any, all, percentage Ann and Bob won’t be home. Being friendly is the easiest way So, Ann won’t be home. to make friends quickly. Alana has a lot of friends. Mikhail came from Italy and All oranges are fruits. All fruits Therefore, Alana must be very Italians are tall, therefore Mikhal grow on trees. Therefore, all friendly. is tall. oranges grow on trees. Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 Example 2: If Sue wins, then Ed will be happy. If Ed is happy, then The sum of three numbers is 64. If the second number is two times the George will be happy. So, if Sue first and the third is four more than the second, then find the numbers. wins, then George will be happy. 1. Given: Let x = 1st number (walang binigay eh) 2x = 2nd number Polya’s Strategy 2x + 4 = 3rd number - Named after George Polya (1887-1985). It is a four-step 2. x + 2x + 2x + 4 = 64 problem solving strategy that is deceptively simple. 3. x + 2x + 2x + 4 = 64 1. Preparation (understand the problem) 5x + 4 = 64 5x = 64 - 4 2. Thinking Time (devise a plan) 5x = 60 3. Insight (carry out plan) 5x/5 = 60/5 4. Verification (look back and review) x = 12 4. 12 + 2(12) + 2(12) + 4 = 64 Example 1: 12 + 24 + 24 + 4 = 64 The sum of two numbers is 30. The first number is twice as large as 64 = 64 the second one, what are the numbers? 1. Given: Let x = 2nd number Example 3: 2x = 1st number The sum of three consecutive integers is 75. Find the numbers. 2. 2x + x = 30 1. Given: Let x = 1st number 3. 2x + x = 30 x + 1 = 2nd number 3x = 30 x + 2 = 3rd number 3x/3 = 30/3 If odd or even, let x = 1st number X = 10 x + 2 = 2nd number 4. Given: Let x = 2nd number - 10 x + 4 = 3rd number 2x = 1st number - 2(10) = 20 2. x + x + 1 + x + 2 = 75 5. Verification: If x = 10 3. x + x + 1 + x + 2 = 75 2x + x = 30 3x + 3 = 75 2(10) + 10 = 30 3x = 75 - 3 20 + 10 = 30 3x = 72 30 = 30 3x/3 = 72/3 x = 24 4. x + x + 1 + x + 2 = 75 24 + 24 + 1 + 24 + 2 = 75 Downloaded by vllchr xxx ([email protected])  lOMoARcPSD|46746737 24 + 25 + 26 = 75 2 + 10 + 3(2) + 10 = 28 75 = 75 12 + 16 = 28 28 = 28 Age Problem: Matt is 6 years more than three times older than his sister. Four years Work Problem: (ALWAYS FRACTION FORM) ago, he was 5 times older as his sister, how old is his sister? Mr. Y can paint a house in 10 days while Mr. Z can paint the house in 1. Let x = sister 15 days. How many days will be required for the painting of the house 3x+6 = Matt if they work together. 2. 4 years ago. 1. Let x = total number of hrs/days working together 3x+6 - 4 = 5 (x-4) 1/10 = Mr. Y 3. 3x+2 = 5x-20 1/15 = Mr. Z 3x-5x = -2-20 2. 1/10 + 1/15 = 1/x -2x = -22 3. [1/10 + 1/15 = 1/x] 30x x = 11 30x/10 + 30x/15 = 30x/x 4. Sister = 11 yrs old 3x + 2x = 30 Matt → 3(11)+6 = 39 yrs old 5x = 30 3x+6-4 = 5(x-4) x = 6 days 3(11) + 2 = 5 (11-4) 4. 1/10 + 1/15 = 1/6 33 + 2 = 5(7) [1/10 + 1/15 = 1/6] 30 35 = 35 30/10 + 30/15 = 30/6 Linda is thrice as old as MJ. Ten years from now, the sum of their ages 3+2=5 will be 28, how old are they now? 5=5 1. Let x = MJ 3x = Linda 2. +10 yrs (x+10) + (3x+10) = 28 3. x + 10 + 3x + 10 = 28 4x + 20 = 28 4x = 28 - 20 4x = 8 x=2 4. MJ = 2 yrs old Linda = 3(2) = 6 yrs old (x + 10) + (3x + 10) = 28 Downloaded by vllchr xxx ([email protected])