Chapter 2 discusses the mathematical language and symbols, emphasizing how this language facilitates communication and precision in expressing mathematical concepts. It explains the differences between expressions and equations, introduces key mathematical symbols, and provides examples on how to represent mathematical statements using symbols. Additionally, it covers set notation, including operations like union and intersection, and illustrates these concepts through various examples.

1. CHAPTER 2:
MATHEMATICAL
LANGUAGE AND
SYMBOLS
AIRA JANE MARAMBA, LPT
INSTRUCTOR

2.  facilitates
communication
and clarifies
meaning
 allows people to
express
themselves and
maintain their
identity.
Language

3. The language of mathematics
makes it easy to express the
kinds of thoughts that
mathematicians like to
express. It is:
precise (able to make very
fine distinctions);
concise (able to say things
briefly);
powerful (able to express
complex thoughts with
relative ease)
Characteristics of the language
of mathematics

4. PHRASE
A GROUP OF WORDS THAT EXPRESSES A CONCEPT
SENTENCE
A GROUP OF WORDS THAT ARE PUT TOGETHER
TO MEAN SOMETHING
ENGLISH

5. MATHEMATICS
EXPRESSION
A group of number or variable with or
without mathematical operation
EQUATION
a group of number or variable
with or without mathematical operation
separated by an equal sign

6. EXPRESSION VS EQUATION
Sum of two numbers
Expression
𝑥 + 𝑦
Sum of two numbers is 8.
Equation
𝑥 + 𝑦 = 8

7. SIGE NGA TRY NYO TO
1. 5
2. 2 + 3
3. x + 3 y = 4xy
4. ½
5. The product of two numbers
6. The sum of three integers is greater than 11.
7. (6 − 2) + 1
8. 𝑥 = 1

8. Symbol Symbol Name Meaning / definition Example
= equals sign equality 5 = 2+3
5 is equal to 2+3
≠ not equal sign inequality 5 ≠ 4
5 is not equal to 4
≈ approximately equal approximation sin(0.01) ≈ 0.01,
x ≈ y means x is approximately
equal to y
> strict inequality greater than 5 > 4
5 is greater than 4
< strict inequality less than 4 < 5
4 is less than 5
≥ inequality greater than or equal to 5 ≥ 4,
x ≥ y means x is greater than or
equal to y
≤ inequality less than or equal to 4 ≤ 5,
x ≤ y means x is less than or equal
to y
MATHEMATICAL SYMBOLS

9. Translate the
following to
mathematical
expressions
/equations.

10. English phrase/sentence
Product of two numbers
Three more than twice a number
Two less than half a number is 15.
The sum of three distinct numbers
is at least 10.
He owns at most eight cars.
The price of the house increased
by 8%.
Each kid gets one-eighth of the
cake.
Mathematical symbols
𝐴 × 𝐵 or 𝐴𝐵
2𝑥 + 3
1
𝑦 − 2 = 15
2
𝑥 + 𝑦 + 𝑧 ≥ 10
𝐶 ≤ 8
𝑃𝑛𝑒𝑤 = 𝑃𝑜𝑙𝑑 +0.08 𝑃𝑜𝑙𝑑
1
𝐾 =
8
𝐶

11. SETS

12. SETS
COLLECTION OF
OBJECTS, CALLED AS
ELEMENTS

13. 1
2
3
4
5
S
𝑺 = 𝟏, 𝟐,
𝟑, 𝟒, 𝟓
ROSTER METHOD
SET NOTATION

14. 𝑺 = 𝟏, 𝟐, 𝟑,
𝟒, 𝟓
1 ∈ 𝑆 means “1 is an element of set 𝑆”
while
6 ∉ 𝑆 means “6 is NOT an element of set 𝑆”
Set Notation

15. 𝑺 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, …
𝑆 also contains 6,7,8, and so on – all
positive integers
SET NOTATION

16. 𝑻 = … , −𝟑, −𝟐, −𝟏
𝑇 also contains -4,-5,-6, and so on – all negative
integers
SET NOTATION

17. 𝒁 = … , − ,
𝟐 − ,
𝟏 ,
𝟎 ,
𝟏 ,
𝟐
…
𝑍 also contains all integers
SET NOTATION

18. WHAT IF I WANT TO KNOW THE SET CONTAINING ALL REAL
NUMBERS BETWEEN 0 AND 1 (INCLUDING 0 AND 1)?
𝑆 = 𝑥 | 𝑥 ≥ 0 𝐴𝑁𝐷 𝑥 ≤ 1
“such that”
“𝑆 contains all 𝑥’s such that 𝑥 is greater than or
equal to 0 AND 𝑥 is less than or equal to 1”
Set-builder
notation

19. Set of natural numbers ℕ = 1, 2, 3, 4, 5, …
Set of integers ℤ = … , −2, −1, 0, 1, 2, …
SOME KNOWN SETS
Empty set ∅ or

20. Describe each set.
• {Sunday, Monday, Tuesday, Wednesday, Thursday,
Friday, Saturday}
• {A, E, I, O, U}
• {Mercury, Venus, Earth, Mars}
• {2, 4, 6, 8, 10}
• {2, 4, 6, 8, 10, …}
• {2, 3, 5, 7, 11, 13}
USING SETS

21. Write in set notation.
• Months with 31 days
• Colors of the rainbow
• Dog breeds that lay egg
USING SETS

22. SUBSET
Set
Subset
1,2,3 ⊆ {1,2,3,4,5}
NOTATION:
{1,2,3} is a subset of {1,2,3,4,5}

23. SUBSET (EXAMPLES AND NONEXAMPLES)
{1,2,3,4,5} is a subset of {1,2,3,4,5}.
{1,2} is a proper subset of {1,2,3,4,5}.
{6,7} is not a subset of {1,2,3,4,5}.
{1,3,6} is not a subset of {1,2,3,4,5}.
The empty set, ∅, is a subset of
1,2,3,4,5 .

24. COMPLEMENT OF A SET
NOTATIONS: 𝐴 – any set
𝐴′ – the complement of
set 𝐴
�
�
𝐴
′
The set 𝐴′
contains elements
in the universal set
which are not
contained in set
𝐴.

25. COMPLEMENT OF A SET (EXAMPLES)
Universal set → 𝐔 = {1,2,3,4,5,6,7,8,9,0}
If 𝐴 = 1,4,5,6 , then 𝐴′ =
{2,3,7,8,9,0}.
If 𝐵 = {1,2,3}, then 𝐵′ =
{4,5,6,7,8,9,0}.
If 𝐶 = {0,1,2,3,4,5,6,7,8,9}, then 𝐶′ = ∅

26. Set operation (Union)
THE UNION OF SETS A AND B, DENOTED BY U,
IS THE SET THAT CONTAINS ALL THE ELEMENTS
THAT BELONG TO A OR TO B OR TO BOTH.
A  B  x x  A or
x  B
U
A B
A 
B

27. SET OPERATION (UNION)
�
�
�
�
Union of 𝐴
and 𝐵
𝑨 ∪ 𝑩
EXAMPLE:
Let 𝐴 =
{1,3,4,5}
𝐵 =
{3,4,7,8}.
Then
𝐴 ∪ 𝐵 =
{1,3,4,5,7,8}

28. Set operation (Intersection)
The intersection of sets A and B, denoted by ∩,
Is the set of elements common to both A and B.
A  B  x x  A and x  B

29. SET OPERATION (INTERSECTION)
EXAMPLE:
Let 𝐴 =
{1,3,4,5}
𝐵 =
{3,4,7,8}.
Then
𝐴 ∩ 𝐵 =
{3,4}
Intersection of 𝐴
and 𝐵
𝑨 ∩ 𝑩
𝐴
𝐵

30. LET U = {1,2,3,4,5,6,7,8,9,10,11,12}
A = 1,3,5,7,9,10
B = {1,2,3,5,7}
C = {2,4,6,7,8}
What is A∩(B ∪ C)’?

31. NEXT TOPIC:
SETS OF FUNCTIONS AND
RELATIONS & BINARY
OPERATIONS

32. THANK YOU

33. SHORT QUIZ

34. If the universal set is given by S={1,2,3,4,5,6}
Where:
A={1,2}
B={2,4,5}
C={1,5,6}
Find the following sets: