MMW

2. Mathematics in the Modern World
LANGUAGE OF SETS

3. Definition of Language
• Language is a systematic way of
communication with other people
use of sounds or conventions
symbols.

4. Importance of Language
• Language was invented to communicate
ideas to others.
• The language if mathematics was
designed:
 Numbers
 Sets
 Functions
 Perform operations

5. Symbols commonly used in
Mathematics
1. The ten digits: 0, 1, 2, … 9
2. Operations: +, –, x, ÷
3. Sets: , , ,
∩ ∪ ⊆ ⊇
4. Variables: a, b, c, x and y
5. Special symbols =, <, >, , , π
≤ ≥

7. The Grammar of Mathematics
• The mathematical notation used for
formulas has its own grammar, not
dependent on a specific natural
language, but shared internationally by
mathematicians regardless of their
mother tongues.

8. SETS
A set is a well-defined collection of distinct objects.
• It is usually represented by capital letters.
• The objects of a set are separated by commas.
• The objects that belong in a set are the elements, or
members of the set.
• It can be represented by listing its elements between
braces.
• A set is said to be well – defined if the elements is a set
are specifically listed.
A = {a, e, i, o, u}
B = {set of plane figures}
C = {Ca, Au, Ag}

9. Examples.
1. The set of all positive number less than 8.
2. The set of even numbers divisible by 3 and less than
50.
3. The set of tall people.
4. The set of months with 30 days.
5. The set of nice cars.
– The statement is a set.
– The statement is a set
– The statement is not a set.
– The statement is a set.
– The statement is not a set.

10. Notation
• If S is a set, the notation
 x ∈ S means that x is an element of S
 x ∉ S means that x is not an element of S
• A variation of notation is used to describe a very
large set
 {1, 2, 3, … 100} refer to set of all integers from 1 to
100.
 {1, 2, 3, …} refer to set of all positive integers.
• The symbol … is called an ellipses and is read “and
so forth”

11. Cartesian Sets of numbers
1. N = {1, 2, 3, …} = the set of natural numbers.
2. W = {0, 1, 2, 3, …} = the set of whole numbers
3. Z = {-3, -2, -1, 0, 1, 2, 3, …} = set of integers
4. Q = the set of rational numbers (terminating or
repeating decimals)
5. Q’ = the set of irrational numbers (non
terminating, non repeating decimals)
6. R = the set of real numbers
7. C = set of complex number

12. TYPES OF SETS

13. Definitions Regarding Sets
• A set is finite if the number of elements is
countable.
EXAMPLES:
A = {even numbers less that 10}
B = {days in a week}
• A set is infinite if the numbers of elements cannot
be counted
EXAMPLES:
A = {even numbers greater than 20}
B = { odd numbers}

14. Equal and Equivalent Sets
Equal sets are set with exactly the same elements
and cardinality.
Example:
A = {c, a, r, e}
B = {r, a, c, e}
Equivalent sets are set with the same number of
elements or cardinality.
Example:
A = {a, e, i, o, u}
B = {1, 2, 3, 4, 5}

15. Joint and Disjoint Sets
Joint sets are set with common elements
(intersections).
Example:
A = {c, a, r, e}
B = {b, e, a, r, s}
Disjoint sets are set with no common elements.
Example:
The set A = {a, b, c} and B = {e, f, g} are disjoint sets,
since no elements is common.

16. Empty Set
The empty set or null set that contains no elements.
The symbol or { } represent the empty set.
∅
Examples:
A = {days start with letter L}
B = {triangle with 4 sides}

17. Subsets
Given two sets A and B, if every element of set A is
also an element of set B, then A is called a subset if B
and we write it as A B.
⊆
Example
Let A = {2, 4, 8} and B = {2, 4, 6, 8, 10}. Since, all the
elements of a set A are contained in set B, then A ⊆
B.

18. Superset
Whenever a set A is a subset of B, we say that B is a
superset of A and we write, B A.
⊇
Example
Let A = {2, 4, 8} and B = {2, 4, 6, 8, 10}. Hence, A B
⊆
and B A
⊇

19. Universal Set
The set of all elements. We will use the letter U to denote
universal set.
Example:
If A = {1, 2, 3} and B = {3, 4, 5,}, then the universal set U is U =
{1, 2, 3, 4, 5}

20. Operation of Sets
Sets can be combine in a number of different ways to
produce another set. There are operations that
involved in a set. The following operations are union,
intersection, complementation, set difference, and
cartesian product.

21. 1. Union of Two Sets
The union of two sets, denoted by the symbol " ", is a
∪
set containing all elements that are in either the first
set, the second set, or both. It combines the elements
from both sets into a single set, with no duplicates.
Example:
Find the union of the following sets:
A = { 2, 3, 4} and B = {3, 4, 5}
A B = {2, 3, 4, 5}
∪

22. 2. Intersection of Two Sets
The intersection of two sets is the set containing all
elements that are common to both original sets. It's
represented by the symbol " ", so A B reads as "A
∩ ∩
intersection B".
Example:
Find the intersection of the following sets:
A = { 1, 2, 3} and B = {1, 2, 4, 5}
A B = {1, 2}
∩

23. 3. Complement of a Set
The complement of a set, denoted as A', or Aᶜ, is the
set containing all elements from the universal set (U)
that are not in the given set A. Essentially, it's
everything "outside" the set A within the context of
the universal set.
Example:
Find the complement of set A = {2, 4, 6, 8} and U = {1,
2, 3, 4, 5, 6, 7, 8, 9, 10}.
A’ = {1, 3, 5, 7, 9, 10}

24. 4. Set Difference
The set difference of two sets A and B, denoted as A –
B or A B, is the set containing all elements that are in
A but not in B. It essentially removes the elements of
B from set A.
Example:
Find the difference.
Given set A = {a, b, c, d, e, f, g} and B = {a, c, f, h, k, u},
find A – B.
A – B = {b, d, e, g}

25. 5. Cartesian Product
The Cartesian product of two sets, A and B, is a new
set formed by creating all possible ordered pairs
where the first element of each pair comes from set A
and the second element comes from set B. It's
denoted as A × B.
Example:
If A = {1, 2} and B = {a, b},
then the Cartesian product A × B is:
A × B = {(1, a), (1, b), (2, a), (2, b)}

26. Venn Diagram
A Venn Diagram is a pictorial representation of the
relationships between two or more sets. We can
represent sets using venn diagrams. In a Venn
Diagram, the sets are represented by shapes; usually
circles or ovals. The elements of a set are labeled
within the circle.