MATHEMATICAL LANGUAGE AND SYMBOLS
CHAPTER 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.
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
𝐴 = 𝜋𝑟 2
∞
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
√4 = 2
°
Degrees
20°
Therefore
A=B
B=A
PERFORM OPERATIONS ON MATHEMATICAL EXPRESSION CORRECTLY
P
Parenthesis
E
Exponents
M
Multiplication
D
Division
A
Add
S
Subtraction

11 − 5

6
2
2
×2−3+1
×2−3+1
 16 − 3 8 − 3
 16 − 3 5
2
2
÷5
 36 × 2 − 3 + 1
 16 − 3(25) ÷ 5
 72 − 3 + 1
 16 − 75 ÷ 5
 73 − 3
 16 − 15
 70
 1
÷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
SPECIFICATIONS OF SET
 There are three main ways to specify a set:
1. List Notation/ Roster Method – by listing all its members
Examples: 1. {1, 12, 24}
2. {a, b, d, m}
2. Predicate Notation / Rule Method – by stating a property od its elements.
Examples: 1. {x|x is a natural number and x<8}
than 8”
means “the set of all x such that x is a natural number and is less
2. {y|y is a student of UC-Banilad and y is older than 25}
3. Recursive Rules – by defining a set of rules which generates or define its members.
Examples: 1. the set of E of even numbers greater than 3:
a. 4∈ E
b. if x ∈ E, then x+2 ∈ E
c. nothings else belongs to E
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}
EQUIVALENT SETS
 Two sets are equivalent if they contain the same number of elements.
Examples:
{1, 2, 3} , {a, b, c} , ∞, 𝜃, 1 , {
,
,
}
All of the given sets are equivalent.
***Note that no two of then are equal but they all have the same number of elements.
UNIVERSAL SET
 A set that contains al elements considered in a particular situation and denoted by 𝑼.
Examples:
a. Suppose we list the digits only.
Then 𝑼 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
b. Suppose we consider the whole numbers
Then 𝑼 = {0, 1, 2, 3, 4, ….} since 𝑼 contains all numbers
SUBSETS
 A set A is called a subset of set B if every element of A is also an element of B. “ A is a subset of B is
written as A⊆ 𝐁.
Examples:
1. A = {7, 9} is a subset of B = {6, 9, 7}
2. D = {10, 8, 6} is a subset of G = {10, 8, 6}
A⊆B
D⊆G
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}
CARDINALITY OF THE SET
 It is the number of distinct elements belongings to a finite set. It is also called the cardinal number of the set A
denoted by 𝑛 A or card A and |A|.
OPERATIONS OF SETS
 Union – is an operation of set A and B in which a set is formed that consists of all the elements
included A or B both denoted by ∪ as A ∪ B.
Examples:
𝑼 = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A ={ 1, 3, 5, 7}
B = {2, 4, 6, 8}
C= {1, 2}
Find the following:
a. A ∪ B
b. A ∪ C
c. (A ∪ B) ∪ {8}
Solution:
a. A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}
b. A ∪ C = {1, 2, 3, 5, 7}
c. (B ∪ C) = {1, 2, 4, 6, 8}
OPERATIONS OF SETS
 Intersection – It is the set containing all elements common to both A and D denoted by ∩.
Examples:
𝑼 = {a, b, c, d, e}
A ={ c, d, e}
B = {a, c, e}
C = {a}
D = {e}
Find the following:
a. B ∩ C
b. A ∩ C
c. (A ∩ B) ∩ D
Solution:
a. B ∩ C = {a}
b. A ∩ C = ∅
c. (A ∩ B) ∩ D = {c, e} ∩ {e}
= {e}
OPERATIONS OF SETS
 Complementation – is an operation on a set that must be performed in reference to a universal set
denoted by 𝑨′ .
Examples:
𝑼 = {a, b, c, d, e}
A ={ c, d, e}
B = {a, c, e}
Find the following:
a. 𝐴′
b. 𝐵′
Solution:
a. 𝐴′ = {a, b}
b. 𝐵′ = {b, d}
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
element Y
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?
4*3 = 3(4) +3
a= 4 b=3
12+ 3
15
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.
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
~ or ¬P
Not / The opposite
Conjunction
p^q
And / Both are True
Disjunction
pvq
Or / One is true, then all is True
Implication
p
Bi-conditional
p
q
If, then / False if q is false and p is true/ True if q is true and
p is false
q
If and only if / True when p and q are both true or false.
TRUTH TABLE
p
q
¬p
Negation
¬q
p^q
pvq
p
q
Negation Conjunction Disjunction 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
p
q
¬p
¬p v q
(¬p v q) ^ p
q
¬p
(¬p v q)
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
(q
¬p)