Chapter I. Nature of Mathematics
Mathematical Language and Symbols
Math 01 – Mathematics in the Modern World
Melvin H. Cayabyab
Department of Mathematics
College of Arts and Sciences
Mathematical Language and Symbols
Math 01 – Mathematics in the Modern World
RECALL
FOUR BASIC CONCEPTS OF MATHEMATICS
1. Sets
2. Relations
3. Functions
4. Binary Operations
Math 01 – Mathematics in the Modern World
FUNCTIONS
Definition
A function is a relation for which each value from the set of
first components of the ordered pairs is associated with
exactly one value from the set of second components of the
ordered pair.
𝑓: 𝐷 → 𝑅 where 𝑓 𝑥 = 𝑦, 𝑥 ∈ 𝐷, 𝑦 ∈ 𝑅.
𝐷 is called the domain of the function and
𝑅 is called the range of the function
Math 01 – Mathematics in the Modern World
FUNCTIONS
Examples:
1. { −1, 1 , 1, 1 , 0, 0 , 2, 4 } is a function
2. { 4, 2 , 4, −2 , 0, 0 , 16, −4 } is not a function
Suppose 𝑓: 0, 1, 2 → {5} where
𝑓 0 =5
𝑓 1 =5
𝑓 2 =5
Graph the function. What if 𝑓: ℝ → ℝ, what is the graph of f?
Math 01 – Mathematics in the Modern World
FUNCTIONS
Figure 1: The graph of 𝑓 𝑥 = 5 where 𝑥 = 0, 1, 2.
Math 01 – Mathematics in the Modern World
Figure 2. The graph of the constant
function 𝑓 𝑥 = 5 over ℝ
FUNCTIONS
Example:
Let 𝑓 ∶ 1,2,3, … , 10 → ℝ where
𝑓
𝑓
𝑓
⋮
𝑓
1 =2
2 =4
3 =6
10 = 20
Can you represent the function 𝑓? Is the graph of the function a set of
discrete points or a continuous line?
Math 01 – Mathematics in the Modern World
FUNCTIONS
Example:
Consider the following function:
𝑓: 0, 1, 2, 3, 4, 5 → ℝ} where
f(0) = 0
f(1) = 1
f(2) = 4
f(3) = 9
f(4) = 16
f(5) = 25
Can you represent the function? Extend this function to the set of real
numbers and sketch the graph. Is it linear?
Math 01 – Mathematics in the Modern World
EXERCISE
Represent the given functions below and sketch their graphs.
1. Let 𝑓: ℝ → ℝ where f(-2) = -3, f(0) = 1, f(5) = 11. [Hint: It
is a linear function.]
2. Let 𝑓: ℝ → ℝ where f(0) = 1, f(1) = 3, f(2) = 9. [Hint: It is a
quadratic function.]
3. Let 𝑓: ℝ → ℝ where f(-1) = -1, f(-1/2) = -0.125, f(0) = 0, f( 2)
= 8, f(3) = 27. What type of function is f?
Math 01 – Mathematics in the Modern World
BINARY OPERATIONS
Definition
Let 𝑆 be a set. A binary operation ∗ on 𝑆 is a function ∗: 𝑆 × 𝑆 → 𝑆
such that ∗ is defined for every pair of elements in 𝑆, and ∗ uniquely
associates each pair of elements of 𝑆 to some element of 𝑆.
To show that ∗ is a binary operation, one needs to satisfy the two conditions below:
i. (Closure). For any two elements 𝑎 and 𝑏 in the set 𝑆, the product 𝑎 ∗ 𝑏 is an element
of 𝑆.
ii. (Uniqueness). Given that 𝑎, 𝑏 ∈ 𝑆 and 𝑎 ∗ 𝑏 = 𝑐, if there exists another element 𝑑 ∈ 𝑆
such that 𝑑 = 𝑎 ∗ 𝑏, then 𝑑 = 𝑐.
Math 01 – Mathematics in the Modern World
BINARY OPERATIONS
Examples:
1. Let 𝑆 = ℝ and ∗ be “+” (usual addition). For 𝑎, 𝑏 ∈ ℝ, 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 ∈
ℝ.
This is a binary operation.
2. Let 𝑆 = ℤ and ∗ be “⋅” (usual multiplication). For 𝑎, 𝑏 ∈ ℤ, 𝑎 ∗ 𝑏 = 𝑎 ⋅
𝑏 ∈ ℤ.
This is a binary operation.
Math 01 – Mathematics in the Modern World
BINARY OPERATIONS
Examples:
3. Let 𝑆 = ℤ and 𝑎 ∗ 𝑏 = max{𝑎, 𝑏}, the largest of a and b.
This is a binary operation.
4. Let 𝑆 = ℤ and 𝑎 ∗ 𝑏 = 𝑎/𝑏.
This is not a binary operation as 𝑎/𝑏 is not defined when 𝑏 = 0 (condition ii)
apart from the fact that 𝑎/𝑏 is not always in ℤ (condition i).
Math 01 – Mathematics in the Modern World
BINARY OPERATIONS
Define a ∗ b = 3a + b, where a, b ∈ ℝ.
Find the following:
1. 8 ∗ 3
2. 3 ∗ 8
3.
1∗2
3∗4
Math 01 – Mathematics in the Modern World
BINARY OPERATIONS
FOR SMALL SETS
For some finite sets, binary operations can be presented using a “multiplication
table”. The resulting entries in the table after performing the operation can provide
relevant information regarding the set.
Example:
Let S={-1, 0, 1} and the operation  be the usual multiplication. Show that  is a
binary operation by means of a multiplication table.
Solution:

-1
0
1
-1
1
0
-1
0
0
0
0
1
-1
0
1
Math 01 – Mathematics in the Modern World
BINARY OPERATIONS
FOR SMALL SETS
Example:
Let ℤ4 = 0, 1, 2, 3 and +4 be the operation that gives the remainder when the
sum of two elements in ℤ4 is divided by 4 (e.g., 2 +4 3 = 1).
Show that +4 is a binary operation on ℤ4 .
Math 01 – Mathematics in the Modern World
PROPERTIES OF
BINARY OPERATIONS
A binary operation may exhibit several properties but not necessarily all of the
following:
Definition
Let ∗ be a binary operation on a set S. Then:
a. ∗ is commutative if for all a, b ∈ S, a ∗ b = b ∗ a.
b. ∗ is associative if for all a, b ∈ S, a ∗ (b ∗ c) = (a ∗ b) ∗ c.
c. An element e of S is an identity for ∗ if for all a ∈ S, a ∗ e = e ∗ a = a.
d.
If e is an identity for ∗ and a ∈ S, then a is invertible if there exists b ∈ S
such that a ∗ b = b ∗ a = e. The element b is called the inverse of a.
Math 01 – Mathematics in the Modern World
EXERCISE
Determine if the binary operation ∗ defined by the table below is commutative.
• Is ∗ associative for these values? 𝑎 ∗ 𝑑 ∗ 𝑐 = 𝑎 ∗ 𝑑 ∗ 𝑐
• Based on the table, what is the identity element?
• What is the inverse of a? b? c? d?
Math 01 – Mathematics in the Modern World
∗
a
b
c
d
a
a
b
c
d
b
b
c
d
a
c
c
d
a
b
d
d
a
b
c
EXERCISE
A binary operation * is defined on the set S = 1, 2, 3, 4 . The table below shows how the
operation is to be performed.
1. Determine whether * is commutative.
2. Is 1 ∗ 4 ∗ 3 = 1 ∗ 4 ∗ 3 ?
3. Find the identity element for the operation *
4. and the inverse of each element in S.
Math 01 – Mathematics in the Modern World
LOGIC
Logic, coming from the Greek word “logos” means an idea, an argument or a reason. It is the
study of correct reasoning. It is the scientific method of judging the truth or falsity of statements.
Definition
A proposition is a statement that is either true or false but not both true
and false under the same condition.
declarative sentence.
Examples:
a) Mayon Volcano is in Naga City.
b) 14 is an even number.
c) (-1, 0) is a point on the y-axis.
Math 01 – Mathematics in the Modern World
Generally, a proposition is a
LOGIC
Some declarative sentences are also propositions, and determining whether
it is true or false depends on the specific value of the variable being used.
Examples:
a) x + 4 = 10.
b) He wrote the book “Mathematics in the Modern World”.
Note: Any sentence which contains a variable is called an open sentence.
Math 01 – Mathematics in the Modern World
LOGIC
Propositions are represented by small letters such as p, q and r. These letter
symbols are called sentential variables.
Examples:
p: The base angles of an isosceles triangle are equal.
q: 1 + 7 ≠ 7.
r: The cubic equation x3 – 8 = 0 has a root equal to 2 of multiplicity 3.
Math 01 – Mathematics in the Modern World
LOGIC
Definition
A compound proposition is a proposition formed by combining two or
more simple statements. It is formed with the use of logical connectives
like “and”, “or”, “if ... then”, and “if...and only if...”.
Math 01 – Mathematics in the Modern World
TYPES OF COMPOUND
PROPOSITION
1. Conjunction
Given any propositions p and q, the compound proposition “p and q”, written p ^ q, is called
the conjunction of p and q or simply a conjunction.
The only way for a conjunction to be true is when all its components are true.
p
q
p^q
T
T
T
T
F
F
F
T
F
F
F
F
Math 01 – Mathematics in the Modern World
Example:
p: June has 31 days.
q: 5 is odd.
p ^ q: June has 31 days and 5 is odd.
Truth Value: False since p is false and q is true.
TYPES OF COMPOUND
PROPOSITION
2. Disjunction
Given any propositions p and q, the compound proposition “p or q” , written as p v q, is called the
disjunction of p and q. In the discussion, the inclusive sense (one, or the other, or both) is used.
The only way for a disjunction to be false is when all its components are false.
p
q
pvq
T
T
T
T
F
T
F
T
T
F
F
F
Math 01 – Mathematics in the Modern World
Example:
p: Taal Volcano is in Laguna.
q: 3 + 2 = 6
p v q: Taal Volcano is in Laguna or 3 + 2 = 5.
Truth Value: True since p is true and q is false.
TYPES OF COMPOUND
PROPOSITION
3. Negation
This is the denial of a statement. It uses phrases such as “not”, “it is not true that” or “it
is false that”. Notations for negation can be any ¬, − or ~.
If p is any proposition, then its negation is denoted by
¬p,
or
−p
or
~p
Examples:
Proposition
1. p: 5 is divisible by 2.
2. q: 4 + 2 = 6.
Math 01 – Mathematics in the Modern World
Truth
Value
False
True
Negation of the Proposition
¬p : 5 is not divisible by 2.
¬q: 4 + 2 ≠ 6.
Truth
Value
True
False
TYPES OF COMPOUND
PROPOSITION
4. Conditional Proposition (Implication)
Given any propositions p and q, the compound proposition “if p, then q ” , written as p → q, is
called a conditional proposition or an implication. p is the antecedent or hypothesis while q is
the consequent or conclusion.
The only way for an implication to be false is when p is true and q is false.
p
q
p→q
T
T
T
Example:
p: 32 = 6
q: 5 is odd.
T
F
F
p → q: If 32 = 6, then 5 is odd.
F
T
T
F
F
T
Math 01 – Mathematics in the Modern World
Truth Value: True since p is false and q is true.
TYPES OF COMPOUND
PROPOSITION
Variants of a Conditional Statement
Condition/Implication : p → q
Converse
: q → p
Inverse
: ¬p → ¬q
Contrapositive
: ¬q → ¬p
Math 01 – Mathematics in the Modern World
TYPES OF COMPOUND
PROPOSITION
5. Biconditional Proposition (Double Implication or Equivalence)
Given any propositions p and q, the compound proposition “p if and only if q”, written as p  q is
called a biconditional proposition or an equivalence proposition.
The only way for biconditional to be false is when p and q have different truth values.
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
T
Math 01 – Mathematics in the Modern World
Examples:
p: 52 = 32 + 42
q: 3 - 5  
p  q: 52 = 32 + 42 if and only if 3 - 5  .
Truth Value: False since p is true and q is false
QUANTIFIERS
Many propositions in mathematics are constructed with the use of the word “all”
or the word “some”. These are called quantifiers. (Feliciano and Uy, 1991).
Definition
Quantifiers are constructs that specify the quantity of specimens in the
domain of discourse that satisfy a formula.
2 Kinds of Quantifiers:
1. Universal Quantifier symbolized by ∀ means “for all, for every, for any”.
2. Existential Quantifier symbolized by  means “for some, there exists”.
Math 01 – Mathematics in the Modern World
QUANTIFIERS
Definition
A statement involving a quantifier is called a quantified statement.
Just like an ordinary statement (proposition), a quantified statement has
also its truth value.
Examples:
1. For all natural numbers x, x + 5 = 10.
2. For every real numbers x and y, x + y = y + x.
3. For some natural numbers x, x - 3 = 5.
4. There exists an integer such that 2x – 7 = 3.
5. For every real numbers x, there exists a real number y such that xy = 1 and yx = 1.
Math 01 – Mathematics in the Modern World
QUANTIFIERS
Definition
Universally quantified statements are denoted by
x P(x).
where P(x) is a propositional function with domain of discourse D.
Math 01 – Mathematics in the Modern World
QUANTIFIERS
Math 01 – Mathematics in the Modern World
QUANTIFIERS
Definition
Existentially quantified statements are denoted by
x P(x).
where P(x) is a propositional function with domain of discourse D.
Math 01 – Mathematics in the Modern World
EXERCISE
Math 01 – Mathematics in the Modern World
NEGATION OF QUANTIFIED
STATEMENTS
Examples:
1. P: All math majors are male.
Notation: ∀x P(x)
Negation: There is at least one math major who is not a male.
2 P: Some teachers in the Department of Mathematics know how to operate a computer.
Notation: x P(x)
Negation: All teachers in the Department of Mathematics do not know how to operate
a computer.
Math 01 – Mathematics in the Modern World