MAT H E MA T IC S IN T H E

MODERN W O R L D
WHAT IS MATHEMATICS ?
• Comes from the ancient word “manthanein” meaning “to learn”
• Greek root word “mathesis” means “knowledge”
• Mathematikos or mathemata means “fond of learning”
• Relies on both logic and creativity
• It is seen as a study of patterns and relations
• It is a language that uses carefully defined terms and symbols
• Generally, it is the art of patterns and connections embedded in nature and in our
environment.
PATTERNS
• Patterns are at the heart of mathematics.
• Are regular, repeated or recurring forms or designs
• Example:
1. Layout of the floor tiles
2. Designs of skyscrapers
3. The way we tie our shoelaces
TYPES OF PATTERNS
• SELF-ORGANIZED PATTERNS – NATURALLY MADE PATTERNS
- EXTENDS TO NON-LIVING WORLD
EXAMPLE:
STRIPES OF A ZEBRA
LIGHTNING

• INVOKED ORGANIZATION PATTERNS – SELF-MADE PATTERNS

EXAMPLE:
WEAVER BIRD
BUILDINGS/HOUSES
FIBONACCI NUMBER
• A Fibonacci number is an integer in the infinite sequence 1, 1, 2, 3, 5, 8, 13, … of which the first
two terms are 1 and 1 and each succeeding term is the sum of the two immediately preceding.
The numbers are named after fibonacci, also known as leonardo of pisa or leonardo pisano.

• An important characteristics of sequence is the fact that the ratio between any number and the
previous one in the series tends towards a well-defined value: 1.628… this is the GOLDEN
RATIO/MEAN that frequently occurs in nature.

•
•FIBONACCI NUMBERS:
1,1,2,3,5,8,13,21,34,55,89,134, …,
•NOTE: THE FIRST NUMBER IN THE
FIBONACCI SEQUENCE IS MARKED AS
THEN UNTIL .
EXAMPLES:
1. FIND
SOLUTION:

= 21 + 34
= 55

=
2. PROVE:

• 3. PROVE:
4. PROVE

5.  IF N IS AN EVEN NUMBER, THEN IS AN ODD NUMBER.

NOTICE THAT THEREFORE THIS STATEMENTS IS FALSE.

 𝟏
𝑭 𝒏 = ¿
√𝟓

•
MATHEMATICAL
LANGUANGE AND
SYMBOLS
LANGUAGE

• IMPORTANT TO UNDERSTAND AND EXPRESS ONE’S IDEAS.

• TRANSMITTER OF INFORMATION AND KNOWLEDGE.
• IT HELPS US TO CONSTRUCT SOCIAL IDENTITY.
• MISUNDERSTANDING OF ONE’S LANGUAGE LEADS TO CONFUSION AND MISCONCEPTIONS.
• MATHEMATICS HAS ITS OWN LANGUAGE.
• SYSTEMATIC WAY OF COMMUNICATING WITH OTHER PEOPLE BY THE USE OF SOUNDS AND
CONVENTIONAL SYMBOLS.

• IT WAS DESIGNED SO WE CAN WRITE ABOUT THINGS (NUMBERS, SETS, FUNCTIONS, ETC.)
• MATHEMATICS LANGUAGE CONSISTS OF STRUCTURAL RULES GOVERNING THE USE OF SYMBOLS
REPRESENTING MATHEMATICAL OBJECTS.
VOCABULARY OF MATHEMATICS

• THE 10 DIGITS:
• SYMBOLS FOR OPERATIONS AND SETS:
• SYMBOLS THAT “STANDS IN” :
• SPECIAL SYMBOLS:
• SET NOTATIONS:
Often (but not always) letters are used in a
mathematical statement. And these letters have
special uses:
  EXAMPLES WHAT THEY USUALLY MEAN

Start of the alphabet: a, b, c, … Constants (fixed values)

From I to n: i, j, k, l, m, n Positive integers (for counting)

End of the alphabet: … , x, y, z Variables (unknowns)

 CHARACTERISTICS OF THE MATHEMATICAL LANGUAGE

•PRECISE (ABLE TO MAKE VERY FINE DISTINCTIONS OF

DEFINITION)
•CONCISE (ABLE TO SAY THINGS BRIEFLY)
•POWERFUL (ABLE TO EXPRESS COMPLEX THOUGHTS WITH
RELATIVE EASE)
Note: To fully understand language we need to know the difference
between English sentences and English expression.
 ENGLISH EXPRESSION / MATHEMATICAL EXPRESSION

• IS A NAME GIVEN TO A MATHEMATICAL OBJECT OF INTEREST.

Example:
IT SIMPLY MEANS, IT IS A WORD/S, PHRASE/S, NAME/S THAT DON’T CONVEY
English Expression
A COMPLETE THOUGHT Mathematical Expression
Dog X (x)(y)
Running along 1 y–2
Linda x+1
 ENGLISH SENTENCE / MATHEMATICAL SENTENCES

•MUST STATE A COMPLETE THOUGHT

Example:
English Sentence Mathematical Sentence

The teacher talks in front of the x = 2 (x)(y) = (y)(x)

class. 1+1=2 y–2=8
A dog barking loudly. x+1=0
Note: In English sentence, there is a certain verb or a linking
verb “is” to be identified and in Mathematical sentence, the “=”
sign is verb
•TRANSLATION
KEYWORDS:
THAN
FROM START WRITING FROM RIGHT TO LEFT
TO

BY start writing from left to right

AND
•EXAMPLE: (LET N BE THE NUMBER)
•A NUMBER INCREASED BY 4
•THE CLUE WORD IS BY, SO START FROM L-R
ANS. N + 4
•A NUMBER LESS THAN 6
6–N
SEVEN LESS THAN A NUMBER IS 15.

17 more than some number is 57.

The sum of 5 and product of 7 and x

The total of 5 and c

The sum of a number and 16 is 23.

FOUR BASIC
CONCEPTS
PREPARED BY: JOAN D. MANGADA
(INSTRUCTOR)
1. A SET IS A COLLECTION OF WELL - DEFINED
OBJECTS, CALLED ELEMENTS.
•NAMES OF SET ARE DESIGNATED USING CAPITAL LETTERS SUCH AS A,
B, …
•TO INDICATE THAT AN ELEMENT BELONGS TO A GIVEN SET, THE SYMBOL IS USED.
•ONE WAY OF WRITING A SET IS BY LISTING ITS ELEMENTS, SEPARATING BY
COMMAS, INCLUDING THIS LISTING WITHIN A PAIR OF BRACES, . THIS WAY OF
REPRESENTING A SET IS CALLED THE LISTING METHOD OR ROSTER METHOD
•NOTE: WHEN WE FORM A SET, THE ELEMENTS WITHIN THE
SET ARE NEVER REPEATED AND THE ELEMENTS CAN APPEAR
IN ANY ORDER.
•THE NOTATION IS AN EXAMPLE OF WRITING THE SET IN SET
BUILDER NOTATION.
• A SET IS SAID TO BE A SUBSET OF ANOTHER SET IF
EVERY ELEMENT OF IS ALSO ELEMENTS OF , IN SYMBOL,

•THE SET IS SAID TO BE EQUIVALENT TO THE

SET (EXACTLY THE SAME NUMBER OF
ELEMENTS)
•THE SET IS SAID TO BE EQUAL TO THE SET , WRITTEN , IF
AND ONLY IF AND (EXACTLY THE SAME ELEMENTS)
•NULL SET/EMPTY SET: A SET WHICH DOES NOT CONTAIN
ANY ELEMENT, IT IS DENOTED BY THE SYMBOL, OR { }.
OPERATIONS ON SET

• UNION: THE UNION OF SETS A AND B, DENOTED BY IS DEFINED AS THE

SET WHOSE ELEMENTS ARE IN A OR IN B IN BOTH AND A AND B.
EXAMPLE:
•INTERSECTION: THE INTERSECTION OF A AND B, DENOTED
BY IS DEFINED AS THE SET WHOSE ELEMENTS ARE COMMON
TO BOTH.

EXAMPLE:
•DIFFERENCE
EXAMPLE:
•COMPLIMENT
EXAMPLE:
PROPERTIES OF SET
OPERATION
COMMUTATIVE PROPERTY
•THE ORDER OF THE SETS THAT SHOW
THE UNION OR INTERSECTION CAN BE
INTERCHANGED.
ASSOCIATIVE PROPERTY

•THE SETS THAT SHOW THE UNION OR

INTERSECTION CAN BE REGROUPED.
IDENTITY PROPERTY
•THE IDENTITY IN THE UNION OF SET IS Ø(NULL
SET), AND THE IDENTITY IN THE INTERSECTION
OF SET IS U(UNIVERSAL SET).
DISTRIBUTIVE PROPERTY
•THE UNION OPERATOR DISTRIBUTES
OVER THE INTERSECTION OPERATOR
AND VICE VERSA.

 
2 AND 3
FUNCTIONS AND
RELATIONS
FUNCTION
• SPECIAL TYPE OF RELATION WHERE IN NO
TWO ABSCISSA/X-COORDINATE IS THE
SAME,
•A RELATION IS A FUNCTION PROVIDED
THERE IS EXACTLY ONE OUTPUT FOR
EACH INPUT.
•IT IS NOT A FUNCTION IF ONE INPUT
HAS MORE THAN ONE OUTPUT.
 Relations
•DOMAIN: X – VALUE
•RANGE: Y – VALUE Functions

•THE VERTICAL LINE TEST: IF IT IS

POSSIBLE FOR A VERTICAL LINE TO
•INTERSECT A GRAPH AT MORE THAN
ONE POINT, THEN THE GRAPH IS
•NOT THE GRAPH OF A FUNCTION.
DETERMINE WHETHER THE RELATION IS A FUNCTION OR NOT.

1. F=
2. G=
3 . H=

4. THE RULE WHICH ASSIGNS TO EACH CELLULAR

UNIT ITS CELULAR PHONE.
 BINARY
OPERTIONS
What is a Binary Operation?
• A rule of combining two values to
create a new value
Properties of Binary Operations
1)Closure Property:
2. Associative Property:

 
3. Identity Property:

 
4. Distributive Property:

5. Commutative Property:
J*C=C*J
 LOGIC
• Study of the principles of correct reasoning
• It helps us to differentiate correct reasoning
from poor reasoning
• It allows us to validate arguments in and out
mathematics
• It illustrates the importance of precision and
conciseness of the language of Mathematics
MATHEMATICAL LANGUAGE
• It is a branch of mathematics with close
connections to computers.
• It includes both the mathematical study of
logic and the applications of formal logic to
other areas of mathematics.
• The importance of logic to mathematics
cannot be overstated
• No conjecture in mathematics is considered
fact until it has been logically proven and truly
valid mathematical analysis is done only within
the rigors of logic.
• Part of the development is the codification of
mathematical logic into symbols.
• With logic symbols and their rules for use, we
can analyze and rewrite complicated logic
statements much like we do with algebraic
statements.
STATEMENT OR PROPOSITION
 
• It is a declarative sentence that is true or false
but not both.
• Propositional Variables such as . are used to
represent propositions.
 EXAMPLE:
1)San Fernando City is the capital of La Union.
ANSWER: PROPOSITION because it has
truth value(TRUE).
2) The girl is beautiful and sexy.
ANSWER: NOT A PROPOSITION
because it neither true or false.
3) Where are you going?
ANSWER: NOT A PROPOSITION
because the sentence is interrogative.
4) Please give me some water.
ANSWER: NOT A PROPOSITION because the
sentence is imperative.
5) Mt. Apo is the tallest mountain in the
Philippines.
ANSWER: PROPOSITION because it is true.
Mt. Apo is the tallest mountain in the Philippines,
having an elevation of 2,956 meters, which offers
a wonderful scene.
6)
ANSWER: PROPOSITION because the sentence
7) if
ANSWER: PROPOSITION because the
sentence has a truth value (FALSE.
8)
ANSWER: NOT A PROPOSITION
because the value of x is not given, hence,
the truth value of the statement cannot be
determined.
REMARKS: If a proposition is
true, then we can say its truth
value is TRUE, and if a
proposition is false, we say its
truth value is FALSE.
 LOGICAL CONNECTIVES
 
• Mathematical statements may be joined by
logical connectives which are used in
combine simple propositions to form
compound statements.
• These connectives are CONJUCTTIONS,
DISCONJUCTIONS, IMPLICATION,
BOCONDITIONAL, and NEGATION.
TRUTH TABLE: It displays the
relationships between the truth values
of propositions. Truth tables are
especially valuable in the
determination of the truth values of
propositions constructed from simpler
propositions.
 Let p and q be propositions.
1) CONJUNCTION: The conjunction of the propositions
p and q is the compound statement “p and q” denoted
as which is, TRUE ONLY when BOTH p and q are
TRUE, otherwise, it is false.
Truth Table for CONJUNCTION
p q 𝒑∧𝒒
T T T
T F F
2) DISCONJUCTION: The disconjunction of the
propositions p and q is the compound statement
“p or q” denoted as which is, FALSE ONLY
when BOTH p and q are FALSE, otherwise, it is
true.
Truth Table for DISCONJUNCTION
p q 𝒑∨𝒒
T T T
T F T
F T T
F F F
3. NEGATION: The negation of the propositions
p is denoted by where the – is the symbol for
“not”. The truth value of the negation is ALWAYS
the reverse of the truth value of the original
statement.
Truth Table for
NEGATION
p ¬𝒑
T F
F T
4) IMPLICATION/ CONDITIONAL: The
implication or conditional of the propositions p
and q is the compound statement “If p, then q”
denoted as which is, FALSE ONLY when p is true
and q is false.
Truth Table for IMPLICATION

p q 𝒑→𝒒

T T T

T F F

F T T

F F T
  

5) BICONDITIONAL: The bi-conditional of the

propositions p and q is the compound statement “p
if only if q” denoted as which is, TRUE ONLY
when BOTH p and q have the SAME TRUTH
VALUES. p
Truth Table for BICONDITIONAL

T Truth Table
T for BICONDITIONAL T

T F F
p q
F T F

F F T
T T T

T F F

F T F

F F T
OPERATOR PRECEDENCE

¬ 1

⋀ 2

∨ 3

→ 4

↔ 5
Compound statement
i: The tour goes to Italy s: The tour goes to Spain
w: We go to Venice
From symbolic for to sentence
1.
= The tour goes to Italy and the tour doesn’t go to Spain
2.
= If we go to Venice, then, the tour goes to Italy
 
Sentence to symbolic
2. The tour goes to Italy and the tour goes to Spain
=
2. The tour doesn’t go to Spain if and only if we go to Venice
=