1. 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. 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.
3. 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 _________.
4. 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
5. THREE OF THE MOST IMPORTANT KINDS OF
SENTENCES IN MATHEMATICS
UNIVERSAL STATEMENT
CONDITIONAL STATEMENT
EXISTENTIAL STATEMENT
6. 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.
7. 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.
8. 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.
9. 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.
10. 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_______.
11. 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 _____.
12. 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_______.
13. 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.
14. 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 ______.
15. 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.
16. 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.
17. 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.
18. 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 .
19. 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.
20. 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)
21. 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 }
22. 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 }
23. 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
24. 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}
25. 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 )’
26. 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}
27. 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
28. 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
29. 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 ,?
30. 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.
31. 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.
32. 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
33. 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.
34. 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.
35. Operations on Propositions
• The main logical connectives
such as conjunctions,
disjunctions, negation,
conditional and biconditional.
36. • 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.
37. • 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
38. 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.
39. • 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
40. 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
41. • 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
42. 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
43. • Negation. The negation of the proposition p is
denoted by ~ p.
• Truth table
p ~ p
T F
F T
44. • 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
45. • 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
46. 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
47. 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
48. 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
49. LOGIC PUZZLE
A logical puzzle is a problem that can be solved
through deductive reasoning.
50. 1. Draw the missing tile from the below matrix.
69. 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.
70. 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.
73. Ans: 21.
Sol.
There are 15 small hexagons and 6 large
ones. The last shape in the bottom row is a
pentagon.
74. 10.Find the relationship between the numbers in the top row
and the numbers in the bottom row, and then determine
the missing number.
75. 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.
84. 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.
89. 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.
91. •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.
92. 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
93. 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
94.
95. 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
96. 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:
97. 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
