1. Mathematics in the Modern World – UNIT 2
Mathematical Language
& Symbols
Department of Mathematics and Physics
University of Santo Tomas
Prepared by Eugenio Cedric T. Corro
and Xandro Alexi A. Nieto
3. What is a language?
Language (n.): a
systematic means of
communicating ideas or
feelings by the use of
conventional symbols,
sounds, or marks having
understood meaning
8. Phrasea group of words that expresses a concept
Sentencea group of words that are put together to
mean something
9. Expressiona group of number or variable
with or without mathematical operation
Equationa group of number or variable
with or without mathematical operation
separated by an equal sign
13. English words to mathematics
English phrase/sentence Mathematical symbols
Product of two numbers 𝐴 × 𝐵 or 𝐴𝐵
Three more than twice a number 2𝑥 + 3
Two less than half a number is 15. 1
2
𝑦 − 2 = 15
The sum of three distinct numbers is at
least 10.
𝑥 + 𝑦 + 𝑧 ≥ 10
He owns at most eight cars. 𝐶 ≤ 8
The price of the house increased by 8%. 𝑃𝑛𝑒𝑤 = 𝑃𝑜𝑙𝑑 + 0.08 𝑃𝑜𝑙𝑑
Each kid gets one-eighth of the cake.
𝐾 =
1
8
𝐶
14. Expression or sentence?
Classify.
(1) The product of two numbers
(2) The sum of three integers is greater than 11.
(3) Half of the sum of 23 and 88
(4) The sum of two numbers is half their product.
(5) 2𝑥 − 3
(6) 𝑥 = 1
(7) 𝑥 +
3𝑦
2
(8) 𝑥 + 2𝑥 + 3𝑥 + 4𝑥 + 5𝑥
15. Characteristics of math language
• Precise
- able to make very fine distinctions
• Concise
- able to say things briefly
• Powerful
- able to express complex thoughts with
relative ease
19. 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
20. 𝑺 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, …
𝑆 also contains 6,7,8, and so on – all positive integers
Set Notation
21. 𝑻 = … , −𝟑, −𝟐, −𝟏
𝑇 also contains -4,-5,-6, and so on – all negative integers
Set Notation
22. 𝒁 = … , −𝟐, −𝟏, 𝟎, 𝟏, 𝟐, …
𝑍 also contains all integers
Set Notation
23. What if I want to know the set containing
ALL real numbers between 0 and 1
(including 0 and 1)?
Set Notation
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
24. What if I want to know the set containing
ALL real numbers between 0 and 1
(including 0 and 1)?
Set Notation
𝑆 = 𝑥 | 𝑥 ≥ 0 ∩ 𝑥 ≤ 1
25. What if I want to know the set containing
ALL real numbers between 0 and 1
(including 0 and 1)?
Set Notation
𝑆 = 𝑥 | 0 ≤ 𝑥 ≤ 1
26. 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
28. Write in set notation.
• Months with 31 days
• Colors of the rainbow
• Dog breeds that lay egg
• Three core Thomasian values
Using sets
29. 1. Explain why is incorrect.
2. Explain why is incorrect.
3. Consider the set . Does this set have
one or two elements? Explain.
2 1,2,3
1 1,2,3
1, 1
31. 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 .
32. Complement of a set
NOTATIONS: 𝐴 – any set
𝐴′ – the complement of set 𝐴
𝐴
𝐴′ The set 𝐴′ contains
elements in the
universal set which
are not contained
in set 𝐴.
33. 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′
= ∅
34. 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
35. Set operation (Union)
𝐴 𝐵
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}
36. 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 =