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- VARIABLES THE LANGUAGE OF SETS THE LANGUAGE OF RELATIONS AND FUNCTIONS VARIABLE: > IS A SYMBOL, COMMONLY AN ALPHABETIC CHARACTER, THAT REPRESENTS A NUMBER, CALLED THE VALUE OF THE VARIABLE, WHICH IS EITHER ARBITRARY, NOT FULLY SPECIFIED, OR UNKNOWN. THE ADVANTAGE OF USING A VARIABLE IS THAT ALLOW YOU TO GIVE A TEMPORARY NAME TO WHAT SEEKING . EXAMPLE: 1. IS THERE A NUMBER X WITH THE PROPERTY THAT 2X + 3 = 𝐱𝟐 ? > IS THERE A NUMBER___WITH THE PROPERTY THAT 2 .___ + 3 = ___𝟐 ?
- 2. Are there with the property that the sum of their squares equals the square of their sum ? Answer: Are there numbers a and b with the property that 𝒂𝟐 + 𝒃𝟐 = (a+ b)𝟐 ? 3. Given any real number, its square is nonnegative. Answer: Given any real number x, 𝒙𝟐 is nonnegative.
- Fill in the blanks to rewrite the following statement: For all real numbers x, if x is nonzero then 𝑥2 𝑖𝑠 positive. A. If a real number is nonzero, then its square_______. B. For all nonzero real numbers x, _______. C. If x _______, then _______. D. The square of any nonzero real number is ______. E. All nonzero real numbers have _________.
- Solution: a. Is positive b. 𝐱𝟐 𝐢𝐬 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 c. Is a nonzero, 𝐱𝟐 𝐢𝐬 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 d. Positive e. Positive squares or squares that are positive
- THREE OF THE MOST IMPORTANT KINDS OF SENTENCES IN MATHEMATICS UNIVERSAL STATEMENT CONDITIONAL STATEMENT EXISTENTIAL STATEMENT
- The Universal Statement- says that a certain property is true for all elements in a set. Example: 1. All positive numbers are greater than zero. 2. All negative numbers are lesser than zero.
- Conditional Statement: Says that if one thing is true then something also has to be true. Example: 1. If 378 is divisible by 18, then 378 is divisible by 6. 2. If there is a typhoon, then crops will be destroyed.
- Existential Statement: Says that there is at least one thing for which the property is true. Example: 1. Either peter is a man or a dog. 2. Either the capital of Agusan del Norte is Butuan City or Cabadbaran City. 3. Either 3 times 2 equals 6 or equals 8.
- Universal Conditional Statements: Universal statements contain some variation of the words “ for all” . Conditional statements contain versions of the words “ if – then”. Universal Conditional Statements: > is a statement that both universal and conditional.
- Example: 1. Every dog is an animal, if browny is a dog, then browny is an animal. 2. All men are mortal, if peter is a man, then peter is mortal. For all real numbers x, if x is greater than 2, then 𝑥2 is greater than 4. a. If a real number is greater than 2, then its square is _____. b. If x ______, then_______.
- Universal Existential Statements: Is a statement that is universal because its first part says that a certain property is true for all objects of a given type and it is existential because its second part asserts the existence of something. Example: 1. Every real number has an additive Inverse. 2. Every pot has a lid. a. All pots _____. b. For all pots P, there is ______. c. For all pots P, there is a Lid L such that _____.
- Solution : a, b, c. a. have lids b. a lid for P c. L is a lid for P. Evaluation: Fill in the blanks to rewrite the following statement: All bottles have cap. a. Every bottle _____. b. For all bottles B, there_______. c. For all bottles B. there is a cap C such that_______.
- Existential Universal Statements: > is a statement that is existential because its part asserts that certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind. Example: 1. Some positive integer is less than or equal to every positive integer. Or : There is a positive integer m that is less than or equal to every positive integer. Or : There is a positive integer m such that every positive integer is greater than or equal to m. Or: There is a positive integer m with the property that for all positive integers n, m < n.
- Fill in the blanks to rewrite the following statement in three different ways: There is a person in my class who is at least as old as every person in my class. a. Some_____ is at least as old as_______. b. There is a person p in my class such that p is ______. c. There is a person p in my class with the property that for every person q in my class, p is ______.
- Solution; a. Person in my class, every person in my class. b. at least as old as every person in my class. c. at least as old as q.
- THE LANGUAGE OF SETS Set: Is a collection or a group of well defined distinct objects. Word set as a formal mathematical term was introduced in 1879 by Georg Cantor ( 1845 – 1918). Examples of sets: 1. The set of counting numbers less than 20. 2. The set of whole numbers less than 10. 3. The set of vowels in the English alphabet. 4. The set of prime numbers less than 19.
- TWO WAYS OF DESCRIBING A SET 1. Roster Method which is done by listing or tabulating the elements of a set. Example; 1. A = 1,2,3,4,5,6,7 2. B = -5,-4,-3,-2,-1 3. C = 1,2,3,4,5… 2. Set – Builder Method which is done by stating or describing the common characteristics of the elements of the set. 1. A = X / X is an integer greater than 0 but less than 8 2. B = X / X is an integer less than 0 but greater than -6 3. C = x /x is a counting number 1. Read as A equals the set of all x’s, such that x is an integer greater than 0 but less than 8.
- Three dots ( …) called ellipsis indicates that there are elements in the set that are not written down. Note that braces are used, not parentheses ( ) or brackets to enclose the elements of a set. Kinds of Sets: 1. Finite sets – have a definite limited number of elements. Example: A = 1,2,3,4,5 2. Infinite sets - Have unlimited number of elements. Example : B = 1,2, 3,5,7,11,… 3. Null set = a set with no element. Also called empty set. symbol Ǿ or .
- 4. Equivalent sets – the number of elements in both sets are equal. Example: A = a,b,c and B = 1, 2 ,3 5. Equal sets – Two sets are equal if both sets have the same elements. A = 5,6,7 and B = 5,6,7 6. Disjoint sets = Two sets do not have a common elements . Example: A = 1,2,3 and B = 4,5 ,6 7. Intersecting sets = Two sets have a common elements. Example: A = 1,2,3,4 and B = 4,5,6,7 8. Universal sets = is a totality of elements. 9. Subsets = every element of A is an element of another set. If A = a,b,c and B = c . Set B is a subset of A.
- CARTESIAN PRODUCT : is product of two sets. “ set A cross set B “. A = 1,2 B = 4,5,6 A X B = ( 1,4) , (1,5) , ( 1,6), (2,4) , ( 2,5) , (2,6)
- Basic Operations on sets: 1. The Union: U The union of A and B: A U B Given: Set A = { 1,2,3,4,5}, Set B = { 2,3,4,5,6,7} A U B = { 1,2,3,4,5,6,7} 2. The intersection: The intersection of A Ռ B Given: Set A = { 1,2,3,4,5}, Set B = { 2,3,4,5,6,7} A Ռ B = { 5 }
- 3. The difference: The difference of A and B : A – B Set A = { 1,2,3,4,5,6,7} Set B = { 2,4,6,8,10} 1. A – B = { 1,3 ,5 ,7} 2. B – A = { 8, 10 }
- 4. Universal set: Set U = { 1,2,3,4,5,6,7,8,9,10} Set A = { 2, 4,6,8} Set B = { 2,4,9} Set C = { 2, 6, 8} Find: AU B, A Ռ B , A – B, B – A
- 5. The complement of a set: is the set of elements found in the universal set, but not found in the given set. ( A’ or A prime). Set U = { 1,2,3,4,5,6,7,8,9,10} Set A = { 2,4,6,8} A’ = { 1,3,5,7,9}
- Enumerate the ff. relationships among the given sets: Given: Set U = { 3,4,5,6,7,8,9,10,11,12,13,14,15} Set A = { 3,5,7,9} , set B = { 4,6,8,10} , set C = { 5,10,15} 1. A U B 8. B’ 15. ( B U C )’ 2. A U C 9. C’ 16. (A Ռ B)’ 3. B U C 10. A – B 17. ( A U B U C)’ 4. A Ռ B 11. A –C 18. A’ UB’ 5. A Ռ C 12. B – C 19. ( B Ռ C)’ 6. B Ռ C 13. ( A U B)’ 20. A Ռ B Ռ C 7. A’ 14. ( A U C )’
- Given the elements of the sets. Set X = { 5,6,7,8,9} SET Y = { 7, 8,11,12} SET Z = { 8,9,10,11} Give the operations involved. 1. { 5,6,7,8,9} 6. { 7,8} 2. {7,8,11,12} 7. { 8} 3. { 8,9,10,11} 8. { 5,6,9} 4. { 8,11} 9. { 11,12} 5. { 8,9} 10. { 9,10}
- THE LANGUAGE OF RELATIONS AND FUNCTIONS FUNCTIONS: A variable Y is said a function of X, if each value of X, there is corresponds exactly one value of Y. A function may be written as “ y is function of x or y” = f(x). Read as y equals f of x. Evaluation of a function: To evaluate a function means to substitute a given value to the variable , then solve for y. 1. Evaluate f(x) = -2x +2 when x = 5 2. Evaluate f(x) = 1 + x 1 + x when x = ¾ 3. 𝐄𝐯𝐚𝐥𝐮𝐚𝐭𝐞 𝐟 𝐱 = 𝐱𝟑 +𝟐𝒙𝟐 + 𝟒𝒙 − 𝟐 when x = 4
- RELATION: A relation from X to Y is a set of ordered pairs ( X,Y). In general , a relation is any set of ordered pairs. Example 1. Given: R = ( 1,4) , ( 2,5 ) , (3, 6) , ( 4, 7 ) Domain: 1,2,3,4 Range: 4,5,6,7
- PROBLEM SOLVING defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. THE TYPES OF REASONING 1. Inductive reasoning 2. deductive reasoning Inductive reasoning – is the process of reaching from specific to a general conclusion . Example 1. Use inductive reasoning to predict the next number. 1.1 given: 3 ,6, 9,12,15 ,?
- 1.2 1 is an odd number. 11 is an odd number. 21 is an odd number. therefore, all number ending with 1 are odd numbers. 1.3 Peter is mortal. John is mortal. therefore, all men are mortal.
- DEDUCTIVE REASONING Is the process of reaching a conclusion by applying general to specific procedures or principles. Example 1: Pick a number. Multiply the number by 8, add 6 to the product , divide the sum by 2 and subtract 3.
- Solution: let x a number: Multiply the number by 8 : 8x Add 6 to the product: 8x + 6 Divide the sum by 2 : 8x + 6 2 = 4x +3 Subtract 3: 4x + 3 – 3 = 4x
- 2. All birds have feathers. Eagle is a bird. therefore, eagle have feathers. 3. All integers are real numbers. 2 is an integer. therefore, 2 is a real number.
- Logic: > is the science and art of correct thinking. Aristotle: is generally regarded as the father of Logic. ( 382 – 322 BC) PROPOSITION: or (Statement) - Is a declarative sentence which is either true or false, but not both.
- Operations on Propositions • The main logical connectives such as conjunctions, disjunctions, negation, conditional and biconditional.
- • George Boole in 1849 was appointed as chairperson of mathematics at Queens College in Cork, Ireland. Boole used symbols p, q and symbols ʌ , v , ~ , , , to represent connectives.
- • Logic connectives and symbols: • Statement connective symbolic form Type of statement • Not p not ~ p negation • p and q and p ʌ q conjunction • p or q or p V q disjunction • If p , then q if … then p q conditional • P if and only if q if and only if p q biconditional
- TRUTH VALUE & TRUTH TABLES • Truth Value of a simple statement is either true ( T) or false (F). • Truth Table is a table that shows the truth value of a compound statement for all possible truth values of its simple statement.
- • Conjunction. The conjunction of the proposition p and q is the compound proposition. Symbolically p ^ q. Truth table p q p ^ q T T T T F F F T F F F F
- EXAMPLES: 1. 2 + 3 = 5 and 3 x 2 = 6. p = t q = t 2. 6 x 7 = 42 and 3 x 6 = 19. p = t q = F 3. 2 x5 = 11 and 8 x 8 = 64. p = F q = t 4. 4 x 5 = 21 and 5 x 5 = 26. p = F q = F
- • Disjunction. p or q symbolically p v q. truth table p q p v q T T T T F T F T T F F F
- Examples: 1. 3 or 5 are prime numbers. p = t q = t 2. 2 x 5 = 10 or 2 x 6 = 13. p = t q = F 3. 3 x5 = 16 or 4 x 6 = 24. p = F q = T 4. 20 x 2 = 30 or 10 x 4 = 50. p = F q = F
- • Negation. The negation of the proposition p is denoted by ~ p. • Truth table p ~ p T F F T
- • Examples: Peter is a boy. ( p) true • Peter is not a boy. ( ~ p ) false Dog is a cat. ( p) false Dog is not a cat. ( ~ p) true
- • Conditional. The conditional of the proposition p and q is the compound proposition if p then q. p q . truth table p q p q T T T T F F F T T F F T
- Examples: 1. IF VENIGAR IS SOUR , THEN SUGAR IS SWEET. p = t q = t 2. IF 2 + 5 = 7 , THEN 5 + 6 = 12 p = t q = F 3. IF 14 – 8 = 10, THEN 20 / 2 = 10 p = F q = t 4. IF 2 X 5 = 8 , THEN 40 / 4 = 20. p = F q = F
- BICONDTIONAL . The biconditional of the proposition p and q is the compound proposition p if and only if q. Symbolically p q. truth table p q p q T T T T F F F T F F F T
- Examples: 1. 3 is an odd number if and only if 4 is an even number. p = t q = t 2. 10 is divisible by 2 if and only if 15 is divisible by 2. p = t q = F 3. 8 is a prime number if and only if 2 x 4 = 8. p = F q = T 4. 8 – 2 = 5 if and only if 4 + 2 = 7. p = F q = F
- LOGIC PUZZLE A logical puzzle is a problem that can be solved through deductive reasoning.
- 1. Draw the missing tile from the below matrix.
- answer: Sol. Each lines contains a heart, spade and club with one of the symbols inverted.
- 2. Feed me and I live, yet give me a drink and I die.
- Answer: FIRE
- 3.
- ANSWER: 19 Explanation : As you move diagonally down, numbers follow the sequence of Prime Numbers.
- 4. What number should replace the question mark?
- •Answer Ans: 17.Sol. It is the sum of the two digits(9 + 8) in the quadrant opposite.
- Answer : 9 : The number at the center of each triangle equals the sum of the lower two numbers minus the top number
- 5. A large volume of water is gushing through a pipe which narrows at the outlet. At which point, A, B, C or D will the water flow fastest?
- Ans: C. Sol. The water flows fastest at the narrowest point.
- 4. How many lines appear below?
- ANSWER: 12 Lines
- 5. What number should replace the question mark?
- Ans: 0. Sol. Looking at lines of numbers from the top : 9×8 = 72; 72×8 = 576; 576×8 = 4608; •Answer
- 6. Which three of the four pieces below can be fitted together to form a perfect square?
- Answer:
- 7. Which does not belong in this sequence?
- Ans: E. Sol. The black dot is moving one corner clockwise at each stage and the white dot is moving one corner anti-clockwise at each stage.
- 8. Complete two eight-letter words, one in each circle, and both reading clockwise. The words are synonyms. You must find the starting points and provide the missing letters.
- Ans: ESCALATE, HEIGHTEN. Sol. ESCALATE, HEIGHTEN.
- 9.How many hexagon (six sided figures) can you find here?
- Ans: 21. Sol. There are 15 small hexagons and 6 large ones. The last shape in the bottom row is a pentagon.
- 10.Find the relationship between the numbers in the top row and the numbers in the bottom row, and then determine the missing number.
- Ans: The missing number is 12. Sol. When you look at the relationship of each set of top and bottom numbers, you will see that 12 is 2 x 6, 24 is 3 x 8, 36 is 4 x 9, 45 is 5 x 9, 60 is 6 x 10, and 84 is 7 x 12.
- 11. What symbol is missing?
- Ans: D. Solution:
- 12. Which is the odd one out.
- Ans: E. Sol. All the others have four twigs on the left side branch and three twigs on the right side. Option E is the other way round.
- 13. What number should replace the question mark?
- Answer: 49. Sol. ( 73 + 25) ÷ 2 = 49.
- 14. What number should replace the question mark?
- Ans: 10. Solution: Each diagonal line of numbers, starting with the top left hand corner number, increases by 1 each time, ie :
- EXCURSION meaning: a short involvement in a new activity. Kenken Puzzles: - Is an arithmetic – based logic puzzle that was invented by the Japanese mathematics teacher Tetsuya Miyamoto in 2004. the noun “ ken” has “knowledge” and “awareness” as synonyms. - Rules for solving a kenken puzzle: - For a 3x3 puzzle, fill in each box ( square) of the grid with one of the numbers 1,2,3. - For a 4x4 puzzle, fill in each box ( square) of the grid with one of the numbers 1,2,3,4.
- 2 1 3 3 1 1 3 2 b a c c a a c b
- 2 1 3 4 3 2 4 1 1 4 2 3 4 3 1 2 D
- Magic Square How To Find A Magic Square Solution? Solving a 3 by 3 Magic Square We will first look at solving a 3 by 3 magic square puzzle. First off, keep in mind that a 3 by 3 square has 3 rows, and 3 columns. When you start your 3 by 3 square with you will either choose or be given a set of nine consecutive numbers start with to fill the nine spaces. Here are the steps: •List the numbers in order from least to greatest on a sheet of paper. •Add all nine of the numbers on your list up to get the total. For example, if the numbers you are using are 1, 2, 3, 4, 5, 6, 7, 8. and 9, you would do the following: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. So 45 is the total. •Divide the total from Step 2 by 3. 45 divided by 3 = 15. This number is the magic number. (The number all rows, columns, and diagonals will add up to.) •Go back to your list of numbers and the number in the very middle of that list will be placed in the center of the magic square.
- 1 2 3 4 5 6 7 8 9
- •x represents the number in the center square so for our example, x is equal to 5. •Place the number x + 3 in the upper right-hand square. •Place the number x + 1 in the upper left-hand square. •Place the number x - 3 in the lower right-hand square. •Place the number x -1 in the lower upper left-hand square. •For the last step, fill in the remaining squares keeping in mind the magic number is 15. So that means all the rows, columns, and diagonals need to add up to 15. You should have no problem figuring where to place the remaining numbers keeping this in mind. The complete magic square is in picture below.
- A Magic Square 4 x 4 can be considered as the King of all the Magic Squares, for its an array of 16 numbers which can be added in 84 ways to get the same Magic Sum. Example 1 As usual we are going to make a magic square with the first natural numbers, the magic sum of first 16 natural numbers is 34. Going by our formula. Natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
- STEP– 1 Draw 4 x 4 square grid and put small ‘dots’ in the boxes along both diagonals. Start counting from the topmost left and fill the un dotted boxes with the respective terms. Although the dotted boxes are counted the term corresponding to that box is not written there and the adjacent box is filled in the order.( Here the crossed cells are not filled even though they are counted with numbers written in it,) STEP– 2 Now start filling from the lower most right cell, the boxes are counted using the same NN towards left. The dotted boxes are then filled with the numbers corresponding to that box. Alternate Method. The same result can be obtained if we fill all the cells in the order with out putting any vacant cells along the diagonals and then reversing the filling in these diagonals. Here the Magic Sum a) Along Rows = 34 b) Along Columns = 34 c) Along Diagonals = 34
- 5x5 Magic Squares This is the basic 5x5 magic square. It uses all the numbers 1-25 and it adds up to 65 in 13 different ways: •All 5 vertical lines add up to 65 •The two diagonals both add up to 65 •Finally you can add up the four corners and the number in the middle to get 65. •All 5 horizontal lines add up to 65
- 1 8 5 7 4 6 3 2 How to lay out a 5x5 Magic Square Have another look at the way the numbers are set out in the original square. It uses all the numbers 1-25, and if you follow the numbers round in order you'll see they appear in this pattern: 1,2,3,4 and 5 are in a diagonal line, which goes off the top and comes back at the bottom, then goes off the right and comes back on the left. Once the first five numbers are in place, there's no empty place to put number 6. The rule is to put the 6 UNDER the 5, and then continue putting numbers in another diagonal line:
- Again you'll see that the numbers 6-10 are in a diagonal which goes round until there's no space for the 11. So the 11 goes under the last number which was 10. If you keep going, you'll fill the whole grid with the numbers 1-25 and make the basic 5x5 magic square. 1 8 5 7 4 6 3 2 9
- 1 8 15 5 7 14 16 4 6 13 10 12 3 11 2 9
- 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9
- STEP 4 8 1 6 3 5 7 4 9 2 17 10 15 12 14 16 13 18 11 STEP 2 STEP 3 6 X 6 MAGIC SQUARE 26 19 24 21 23 25 22 27 20 35 28 STEP 4 33 30 32 34 31 36 29 STEP 1
- 5
- FINAL SOLUTION
- 7 X 7 MAGIC SQUARE:
- 7 X 7 MAGIC SQUARE:
- 8 X8 MAGIC SQUARE:

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