Mathematically the most common form is a combination with the other three elements in a given equation that can then determine how the equation will change over the course and how the other two are represented by each element will change in a particular direction in a particular way that the equation will be different than one or two elements are represented in a different direction in the equation that will change in a different order of motion in the order that the other elements will be different.

1. MATHEMATICAL LANGUAGE AND SYMBOLS
CHAPTER 2

2. THE LANGUAGE, SYMBOLS, SYNTAX AND RULES OF MATHEMATICS
◼ The language of mathematics is the systematic used by mathematicians to communicate mathematical ideas
among themselves.
◼ Mathematics as a language has symbols to express a formula or to represent a constant. It has syntax to make the
expression well-formed to make the characters and symbols clear and valid thar do not violate the rules.

3. Symbol Meaning Example
+ Add 3+7 = 10
- Subtract 10-3 = 7
x Multiply 5x6 = 30
÷ Divide 45 ÷5 = 9
/ Divide 45/5 = 9
π Pi
∞ Infinity ∞ is endless
= Equal 1+1 = 2
≈ Approximately π ≈ 3.14
≠ Not equal to 3 ≠ 4
< ≤ Less than, less than or equal to 2 < 3
> ≥ Greater than, greater than or equal to 5 > 2
√ Square root
° Degrees 20°
Therefore A=B B=A

4. PERFORM OPERATIONS ON MATHEMATICAL EXPRESSION CORRECTLY
◼
P E
M
D
A S
Parenthesis
Exponents
Multiplication
Division
Add
Subtraction

5. THE FOUR BASIC CONCEPTS OF MATHEMATICS
Set
◼ A set is a collection of well-defined objects that
contains no duplicates.
◼ The objects in the set are called elements of the
sets.
◼ To describe a set, we use braces {} and use capital
letters to represent it.
◼ Z = {1, 2, 3, …}
Relation
◼ A relation is a rule that pairs each elements in one
set, called the domain, with one or more elements
from a second set called range.
◼ It create sets of ordered pairs.
Holidays Month and Date
New Year’s Day January 1
Labor Day May 1
Independence Day June 12
Bonifacio Day November 30
Rizal Day December 30

6. SPECIFICATIONS OF SET
◼

7. EQUAL SETS
◼ Two sets are equal if they contain exactly the same elements
Examples:
1. {3, 8, 9} = {9, 8, 3}
2. {6, 7, 7, 7, 7} = {6, 7}
3. {1, 3, 5 , 7} ≠ {3, 5}

8. EQUIVALENT SETS
◼ Two sets are equivalent if they contain the same number of elements.
All of the given sets are equivalent.
***Note that no two of then are equal but they all have the same number of elements.

9. UNIVERSAL SET
◼

10. SUBSETS
◼

11. PROPER SUBSET AND IMPROPER SUBSET
◼ Proper subset is a subset that is not equal to the original set, otherwise improper subset.
Examples:
Given: {3, 5, 7}
Proper subset: {}, {5, 7} , {3, 5} , {3,7}
Improper subset : {3, 5, 7}

12. CARDINALITY OF THE SET
◼

13. OPERATIONS OF SETS
◼

14. OPERATIONS OF SETS
◼

15. OPERATIONS OF SETS
◼

16. THE FOUR BASIC CONCEPTS OF MATHEMATICS
Functions
◼ It is a rule that pairs each elements in one set, called
domain (X) and range (Y).
◼ This means that for each first coordinate, there is
exactly one second coordinate or for every first
elements of X, there corresponds a unique second
elementY
Binary
◼ A binary operation on a set is a calculation involving
two elements of the set to produce another element
of the set.
◼ A new math (binary) operation, using the symbol *, is
defined to be a*b = 3a+b, where a and b are real
numbers.
◼ Examples:
What is 4*3? a= 4 b=3
4*3 = 3(4) +3 12+ 3 15

17. ELEMENTARY LOGIC
◼ According to David W. Kueker, logic is simply defined as the analysis of methods of reasoning. Mathematical Logic
is the study of reasoning as used in mathematics.
◼ In ordinary mathematical English the use of “therefore” customarily indicates that the following statements is a
consequence of what comes before.
Examples:
1. All men are mortal. Luke is a man. Hence, Luke is mortal.
2. All dogs like fish. Cyber is a dog. Hence, Cyber likes fish.

18. LOGICAL OPERATORS / CONNECTIVES
◼ Proposition (statement) is a sentence that is either true or false (without additional information) denoted by P
and Q
◼ The logical connectives are defined by truth tables.
Connectives Symbol Words
Negation Not / The opposite
Conjunction p ^ q And / Both are True
Disjunction p v q Or / One is true, then all is True
Implication p q If, then / False if q is false and p is true/ True if q is true and
p is false
Bi-conditional p q If and only if / True when p and q are both true or false.

19. TRUTH TABLE
p q ¬p
Negation
¬q
Negation
p ^ q
Conjunction
p v q
Disjunction
p q
Implication
p q
Biconditional
T T F F T T T T
T F F T F T F F
F T T F F T T F
F F T T F F T T

20. p q ¬p ¬p v q (¬p v q) ^ p q ¬p (¬p v q) ( q ¬p)
T T F T T F F
T F F F F T F
F T T T F T F
F F T T F T F