3. LEARNING
OBJECTIVES
At the end of this session, you are expected to:
1. describe well defined sets, ,subsets, universal sets, null sets, and
cardinality of sets;
2. illustrate the union and intersection of sets and the difference of
two sets;
3. use Venn diagrams to represent sets, subsets, and set operations;
and
4. solve problems involving sets.
4. EXPLORATION
Go to page 2 of your book. Let’s consider the following list below.
Which does not belong to the group?
1. boat, kalesa, car, bus, airplane
2. carabao, chicken, cow, pig, goat
3. Bulacan, Pampanga, Batangas, Bataan, Tarlac
4. hexagon, quadrilateral, rectangle, rhombus, square
5. meter, centimeter, kilometer, square meter, decimeter
6. 2, 12, 24, 11, 30
5. EXPLORATION
If we removed one particular member which is not part of the group,
we can now describe the characteristics of the new formed group.
For instance.
boat, kalesa, car, bus, airplane
boat, car, bus, airplane
(kinds of transport that use fuel)
T = {kinds of transport that use fuel)
6. EXPLORATION
If we removed one particular member which is not part of the group,
we can now describe the characteristics of the new formed group.
For instance.
carabao, chicken, cow, pig, goat
carabao, cow, pig, goat
(animals with for legs)
A = {animals with four legs)
7. EXPLORATION
If we removed one particular member which is not part of the group,
we can now describe the characteristics of the new formed group.
For instance.
Bulacan, Pampanga, Batangas, Bataan, Tarlac
Bulacan, Pampanga, Bataan, Tarlac
(provinces in Central Luzon)
P = {provinces in Central Luzon)
8. EXPLORATION
If we removed one particular member which is not part of the group,
we can now describe the characteristics of the new formed group.
For instance.
hexagon, quadrilateral, rectangle, rhombus, square
quadrilateral, rectangle, rhombus, square
(shapes with four sides)
S = {shapes with four sides)
9. EXPLORATION
If we removed one particular member which is not part of the group,
we can now describe the characteristics of the new formed group.
For instance.
2, 12, 24, 11, 30
2, 12, 24, 30
(even numbers)
E = {even number)
11. ANSWER THE FOLLOWING
QUESTIONS:
1. How many groups are there?
2. Does each object belong to a group?
3. Is there an object that belongs to
more than one group? Which one?
12. DO YOU HAVE THESE GROUPS?
1. H = {kinds of hat)
2. P = {different polyhedrons}
3. T = {kinds of terrestrial tress}
4. N = {set of positive integers}
5. O = {objects in black and white color}
13. THE GROUPS THAT YOU HAVE JUST CREATED ARE
CALLED
SETS
may be thought of as a well-defined collection of objects.
the groups are called sets for as long as the objects in the
group share a characteristic and are thus, well defined.
the objects in the group are called elements (∈) or
members of the set.
14. WAYS OF DESCRIBING
SETS
1.Roster Notation or Listing Method
describing a set by listing each ∈ inside
the symbol { }.
Ex. T = {boat, car, bus, airplane}
15. WAYS OF DESCRIBING
SETS
2. Verbal Description Method
describing a set in words.
Ex. Set T is the set of transport that
uses fuel.
16. WAYS OF DESCRIBING
SETS
3. Set Builder Notation
lists the rules that determine whether an
object is an ∈ of the set rather than the actual
elements.
Ex.
T = {x | x is a kind of transport that uses fuel}
17. NOTICE THE
DIFFERENCE
1. T = {boat, car, bus, airplane}
2. Set T is the set of transport that uses
fuel.
3. T = {x | x is a kind of transport that
uses fuel}
18. LET’S
PRACTICE
A. Describe the set A = {red, orange,
yellow, green, blue, indigo, violet}
using:
1. Verbal description
2. Rule (Set builder notation)
19. LET’S
PRACTICE
Describe the set A = {red, orange,
yellow, green, blue, indigo, violet} using:
1. Verbal description
Set A is the set of the colors of the
rainbow
20. LET’S
PRACTICE
Describe the set A = {red, orange,
yellow, green, blue, indigo, violet} using:
2. Rule (Set builder notation)
A = {x|x is a color of the rainbow}
21. LET’S
PRACTICE
B. Write the elements of
E = {x|x is an integer less than 5 but
greater than - 2}
E = {4, 3, 2, 1, 0, -1}
22. LET’S EXPLORE FURTHER
In the given set below, how many
elements were listed?
E = {4, 3, 2, 1, 0, -1}
Ans: There are 6 elements listed
We can say that n(E) = 6
Cardinality of a set
23. LET’S EXPLORE
FURTHER
EQUAL SETS
Two sets that contain exactly
the same ∈.
Ex.
A = {r, a, i, l}
B = {l, i, a, r}
EQUIVALENT SETS
Two sets that contain equal
number of ∈.
Ex.
A = {1, 2, 3, 4}
B = { J, U, N, E}
Set A is
equivalent to
set B
(A ≈ B)
Set A is equal
to set B
(A = B)
24. LET’S
PRACTICE
A. Identify whether each set is ≈ or =.
A = {r, e, a, d}
B = {x|x is a letter in laptop}
Set C is the components of MAPEH
D = {d, e, a, r}
E = {1, 2, 3, 4, 5}
25. LET’S EXPLORE
FURTHER
UNIVERSAL SETS
A universal set (usually
denoted by U) is a set which
has elements of all the related
sets, without any repetition of
elements.
Ex. If A = {1,2,3} and B = {a,b,c},
then the universal set associated
with these two sets is given by
U = {1,2,3,a,b,c}
VENN DIAGRAM
U
A
1, 2, 3
B
a, b, c
26. LET’S EXPLORE
FURTHER
UNIVERSAL SETS
Try this:
If A = {2,3, 5, 9} and B = {1, 6, 7},
determine its universal set (U)
U = {1, 2, 3, 5, 6, 7, 9}
VENN DIAGRAM
U
A
2, 3, 5,
9
B
1, 6, 7
27. LET’S EXPLORE
FURTHER
SUBSET & PROPER
SUBSET
a set of which all the elements are
contained in another set.
Set A is a subset of B (A ⊆ B) if and
only if every element in A is also an
element in B.
Ex. If A = {1,5,7} and B = {1, 2, 3, 4, 5,
6, 7}, then A ⊆ B
If there is at least one element of set
B that is not an member of set A, we
call set A as the proper subset of B
(A ⊂ B)
VENN DIAGRAM
U
B
2, 3, 4,
6
A
1, 5, 7
30. LET’S
PRACTICE
List all the possible subsets of set given.
1. A = {m, a, t, h}
Zero at a
time
One at a
time
Two at a time Three at a
time
Four at a
time
31. LET’S
PRACTICE
List all the possible subsets of set given.
1. A = {m, a, t, h}
Which of the following subsets given are ⊂?
Zero at a
time
One at a
time
Two at a time Three at a
time
Four at a
time
{ } {m} {m, a} {m, a, t} {m, a, t, h}
{a} {m, t} {m, t, h}
{ t } {m, h} {a, t, h}
{h} {a, t)
{a, h}
{t, h}
32. OPERATIONS ON SETS
INTERSECTION OF SET
The intersection of sets A and
B (A ∩ B) is a set of elements
that are members of both A
and B.
Ex.
A = {1, 2, 4, 6, 8}
B = {2, 3, 4, 5, 6}
A ∩ B = {2, 4, 6}
VENN DIAGRAM
A ∩ B
33. OPERATIONS ON SETS
UNION OF SETS
The union of two sets (A ⋃ B)
contains all the elements
contained in either set (or
both sets).
Ex.
A = {1, 2, 4, 6, 8}
B = {2, 3, 4, 5, 6}
A ⋃ B = {1, 2, 3, 4, 5, 6, 8}
VENN DIAGRAM
Shaded region represents A ⋃ B
34. OPERATIONS ON SETS
DIFFERENCE OF TWO
SETS
The difference of two sets,
written A - B is the set of all
elements of A that are not
elements of B.
Ex.
A = {1, 2, 4, 6, 8}
B = {2, 3, 4, 5, 6}
A – B = {1, 8}
VENN DIAGRAM
Shaded region represents A – B
U
35. OPERATIONS ON SETS
COMPLIMENT OF A SET
The complement of a set (A’) is
the set that includes all the
elements of the universal set (U)
that are not present in the given
set.
Ex.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 4, 6, 8}
B = {2, 3, 4, 5, 6}
A’ = {3, 5, 7, 9}
VENN DIAGRAM
Shaded regions represents A’
U
