The document discusses mathematical language, symbols, syntax, and rules used by mathematicians to communicate ideas. It covers concepts such as sets, relations, functions, binary operations, and logical reasoning. The content also includes examples of mathematical operations and definitions that illustrate these ideas.

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