Mathematics

1. Mathematics
in the Modern
World

2. Chapter 2:
Mathematical
Language and
Symbols

3. 2.1 Mathematics as a Language

4. What is language?
Language is a systematic means of
communicating ideas or feelings by the use of
conventionalized signs, sounds, gestures, or
marks having understood meanings.
Merriam-Webster dictionary

5. According to Dr. Burns, “the language of
mathematics makes it easy to express the kinds of
thoughts that mathematicians like to express.
It is:
1.precise (able to make very fine distinctions);
2.concise (able to say things brief);
3.powerful (able to express complex thoughts
with relative ease).”

6. Some Classification of Symbols
1. Numbers
A number is a mathematical object used to
count, quantify, and label another object. These
include the elements of the set of real numbers
(ℝ), rational numbers (ℚ), irrational numbers (ℚ’),
integers (ℤ), and natural numbers (ℕ).

7. Some Classification of Symbols
2. Operation Symbols include addition (+),
subtraction (-), multiplication (x or ), division (
or /) , and exponentiation (𝑥𝑛
), where x is the base
and n is the exponent.

8. Some Classification of Symbols
3. Relation Symbols include greater than or equal
( ) , less than or equal (), equal ( ), not equal
( ), similar (), approximately equal (), and
congruent (). Congruent figures are the same
shape and size. Similar figures are the same
shape, but not necessarily the same size. On the
other hand, two quantities are approximately
equal when they are close enough in value so the
difference is insignificant in practical terms.

9. Some Classification of Symbols
4. Grouping Symbols include parentheses ( ),
curly brackets or braces { }, or square brackets [ ].
5. Variables are another form of mathematical
symbol. These are used when quantities take
different values. These usually include letters of
the alphabet.

10. Some Classification of Symbols
6. Set theory symbols these are those used in the
study of sets. These include subset (  ), union (),
intersection (), element (), not element (), and
empty set (  ).
7. Logic symbols include implies (),
equivalent (), and (), or (), for all (), there
exists (), and therefore ().

11. Some Classification of Symbols
8. Statistical symbols include sample mean (𝑥),
population mean (), median (𝑥), population
standard deviation (), summation (  ) and
factorial (n!), among others.

12. Mathematical Expression and
Mathematical Sentence
A mathematical expression (analog of a
‘noun’) defined as a mathematical phrase that
comprises a combination of symbols that can
designate numbers (constants), variables,
operations, symbols of grouping and other
punctuation. However, this does not state a
complete thought.

13. Mathematical Expression and
Mathematical Sentence
A mathematical sentence makes a
statement about two expressions. The two
expressions either use numbers, variables, or a
combination of both. It uses symbols or words
like equals, greater than, or less than and it
states a complete thought.

14. Types of Sentences
An open sentence is a sentence that uses variables;
thus it is not known whether or not the
mathematical sentence is true or false.
A closed sentence, on the other hand, is a
mathematical sentence that is known to be
either true or false.

15. Example
The following are mathematical sentences. Label each
of the following as open or closed. For those closed
sentences, identify if it is true or false.
1.10 is an odd number. Answer: Closed - false
2.4 + 5x = 9 Answer: Open
3.10 - 1 = 7 + 2 Answer: Closed - true
4.6 - x = 5 Answer: Open
5.The square root of 4x is 2. Answer: Open

16. Translating Phrases to Mathematical
Expressions or Sentences
Addition (+) Subtraction (−)
Multiplication
(×)
Division (÷)
combined with
plus
the sum of
increased by
total
more than
added to
minus
the difference of
decreased by
fewer than
less than
subtracted from
less
take away
twice (times 2)
thrice (times 3)
squared
cubed
times
the product of
multiplied by
of
divided by
the quotient of
half of
a third of
ratio
shared equally

17. Translating Phrases to Mathematical
Expressions or Sentences
Equal ( = )
Less than or
equal (  )
Greater than or
equal (  )
Equals
Is
Is the same
as
Yields
amount to
at most
not greater than
at least
not less than

18. Example
2n +4 = 14
•Two times a number increased by 4 is 14. Answer: 2n + 4 = 14
•Ten more than thrice a number is at least 12. Answer: 3n + 10  12
•The sum of two consecutive integers is 25. Answer: n + (n +1) = 25
•Subtract 3x from 10xy. Answer: 10xy - 3x
•Ten more than four times a number less than six. Answer: 6 – (4x + 10)
•Ten more than four times a number is less than six. Answer: 4x + 10 < 6
•Nine less a number n Answer: 9 - n

19. 2.2 Four Basic Concepts: Set,
Relation, Function and Binary
Operation

20. Sets
A set is a well-defined collection of distinct objects. The
objects in sets can be anything: numbers, letters, movies,
people, animals, etc. Each object belonging to a set is called
the element or member of the set. For example, the set 𝐶 of
counting numbers less than 4 has numbers 1, 2 and 3 as the
elements. We use the notation “∈” to indicate that a
specific element belongs to a set; otherwise, we use “∉”.
Thus, we write 1 ∈ 𝐶 and 0 ∉ 𝐶 to mean that 1 is an
element of 𝐶 and 0 is not an element of 𝐶, respectively.

21. There are two ways of specifying a set,
namely, roster method and rule method. In
the roster method, the elements of the set are
enumerated, separated by a comma (,), and
enclosed in a pair of braces ({ }). In the rule
method, a phrase is used to describe all the
elements in the set.

22. Definition of terms
The set with no elements is called the empty set or
null set and is denoted by ∅ or { }.
The set with only one element is called the
singleton set.
If a set contains all the elements under
consideration, then it is called a universal set,
denoted by 𝑼.
A set is finite if it consists of a finite number of
elements; otherwise, it is infinite.

23. Definition of terms
Two sets, say 𝐴 and 𝐵, are said to be equal, written
𝐴 = 𝐵, if 𝐴 and 𝐵 have exactly the same elements.
If 𝐴 and 𝐵 have the same number of elements,
then we say that 𝐴 and 𝐵 are equivalent sets.
A set 𝐴 is called a subset of a set 𝐵, written 𝐴 ⊆ 𝐵, if
and only if every element of 𝐴 is also an element of 𝐵.
If 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then we say that 𝐴 is a proper
subset of 𝐵, and write 𝐴 ⊂ 𝐵.

24. Remarks
i. The null set is a proper subset of every set.
ii.Any set is a subset of itself.
iii.A set with 𝑛 elements has a total of 2𝑛
subsets.

25. Operations on Sets
Let 𝐴 and 𝐵 be two arbitrary sets. The union of 𝐴
and 𝐵, written 𝐴 ∪ 𝐵, is the set containing all the
elements which belong to either 𝐴 or 𝐵 or to both.
The intersection of 𝐴 and 𝐵, written 𝐴 𝐵 , is the set
of all elements which are common to both 𝐴 and 𝐵.
The complement of a set 𝐴, written 𝐴′
, is the set of all
elements which are in the universal set 𝑈 but not in
𝐴.

26. Example
Consider the sets 𝑈 = {1, 2, 3, 4, 5}, 𝐴 = 1, 5 , and
𝐵 = {2, 3, 5}. Then
1.𝐴 ∪ 𝐵 = 1, 5 ∪ 2, 3, 5 = {1, 2, 3, 5}
2.𝐴 ∩ 𝐵 = 1, 5 ∩ 2, 3, 5 = {5}
3.𝐵′
= 2, 3, 5 ′
= {1, 4}
4.𝐴 ∪ 𝐵′
= 1,5 ∪ 1,4 = {1, 4, 5}

27. Let 𝐴 and 𝐵 be two non-empty sets. The
Cartesian product of sets 𝐴 and 𝐵, denoted by 𝐴 ×
𝐵, is the set of all ordered pairs (𝑎, 𝑏) where 𝑎 ∈ 𝐴
and 𝑏 ∈ 𝐵.
Example
Consider again the sets in the previous example, 𝑈 =
{1, 2, 3,4, 5}, 𝐴 = 1, 5 , and 𝐵 = {2, 3, 5}. We have
1.𝐴 × 𝐵 = { 1,2 , 1,3 , 1,5 , 5,2 , 5,3 , (5,5)}
2.𝐴 × 𝐴 = { 1,1 , 1,5 , 5,1 , 5,5 }
3.𝐵 × 𝐴 = { 2,1 , 2,5 , 3,1 , 3,5 , 5,1 , 5,5 }

28. Relations and Functions
Intuitively, a ‘relation’ is just a relationship
between sets of information. The couple pairing
and the pairing of students’ names and the
courses taken are examples of a relation. In
mathematics, a relation 𝑅 from set 𝑋 to set 𝑌 is a
subset of 𝑋 × 𝑌. If (𝑥, 𝑦) ∈ 𝑅, then we say that 𝑥
is related to 𝑦 (or 𝑦 is in relation with 𝑥).

29. function
𝒇
Illustration:

30. The domain of the relation 𝑅, denoted by 𝐷(𝑅), is
the set of all first coordinates in the ordered pairs
which belong to 𝑅. That is,
𝐷 𝑅 = 𝑥: 𝑥 ∈ 𝑋, 𝑥, 𝑦 ∈ 𝑅 .
The image of the relation 𝑅, denoted by
𝐼(𝑅), is the set of all second coordinates in the
ordered pairs in 𝑅. That is,
𝐼 𝑅 = 𝑦: 𝑦 ∈ 𝑌, 𝑥, 𝑦 ∈ 𝑅 .

31. 𝑥1
𝑥2
𝑥3
𝑦1
𝑦2
𝑦3
Fig 1. Mapping diagram of
𝑓: 𝑋 → 𝑌
𝑿 𝒀
𝒇

32. Binary Operations
Let 𝑆 be a non-empty set. A binary operation
∗ on 𝑆 is a function from 𝑆 × 𝑆 into 𝑆 such that
for 𝑥, 𝑦 ∈ 𝑆, we have 𝑥 ∗ 𝑦 for ∗ (𝑥, 𝑦). Note that
the image of ∗ is a subset of 𝑆. Thus, we say that
𝑆 is closed under ∗.

33. Example
1.The usual addition (+) , subtraction (−) and
multiplication (∙) are binary operations on the set ℝ
of real numbers.
2.Subtraction(−) and division (÷) are not binary
operations on the set ℕ since 1 − 2 ∉ ℕ and 2 ÷ 3 ∉
ℕ.
3.Let 𝑃 be the set of all sets. The union ∪ and
intersection ∩ of sets are binary operations on 𝑃.

34. Properties of Binary Operations
1.Commutative property
A binary operation is commutative, if ∀ 𝑥, 𝑦 ∈
𝑆, 𝑥 ∗ 𝑦 = 𝑦 ∗ 𝑥.
2.Associative property
A binary operation * on 𝑆 is associative, if
∀ 𝑥, 𝑦, 𝑧 ∈ 𝑆, 𝑥 ∗ 𝑦 ∗ 𝑧 = 𝑥 ∗ 𝑦 ∗ 𝑧 .