Prerequisite Concepts: whole number
About the lesson:
Objectives:
1.
I 1. this lesson, you are expected to:
a. Well-defined sets
b. Subsets
c. Universal set, and;
d. The null set
2.2. Use Venn Diagrams to represent sets and subsets
Lesson Proper:
A
1. Activity
Below are some objects. Group them as you see and label each group
Answer the following questions:
How many groups are there?
Does each object belong to a group?
Is there an object that belongs to more than one group?
The groups are called sets for as long as the objects share a characteristic and are thus well-defined.
Problem: consider consisting of whole numbers from 1 to 200. Let this be set U. form smaller sets consisting of elements of U that share different characteristics for example. Let E be the set of all even numbers from 1 to 200.
Can you form three more such sets? How many elements are there in each of these sets? Do any of these sets have any elements in common?
Did you think of a set with no elements?
Important things to remember
The following are terms you must remember from this point on.
1. A set is a well-defined group of objects called elements that share a common characteristic, for example, 3 of the objects above belong to the set of head covering or simply hats.
2. The set F is a subset of set A if all elements of set F also an element of set A. for example, the even numbers, 2,4 and 12 all belong to the set whole numbers. Therefore the even numbers from a subset of the set of whole numbers. F is a proper subset of A if F does not contain all elements of A.
3. The universal set U is the set that contains all objects under consideration.
4. The null set Ø is an empty set. The null set is a subset of any set.
5. The cardinality of a set A is the number of elements in A.
Notations and symbols
In this section, you will learn some of the notations and symbols pertaining to sets.
- The uppercase letter will be used to name sets and lowercase letters will be used to refer to any element of the set. For example, let H be the set be the set of all objects on page 1 that cover or protect the head we write. H={Ladies' hat, baseball cap, hard hat} This is the listing or roster method of writing the elements of a set. Another way of writing the elements of a set with the use of a descriptor. This is the rule method. For example. H = {x|x covers and protects the head.} This is read as "the set H contains the element x such that x covers and protects the head."
- The symbol ∅ or {} will be used to refer an empty set.
- If F is a subset of A, then we write F ⊆ A. We also say that A contains the set F and we write as A ⊇ F. If F is a proper subset of A, then we write A ⊂ F.
- The cardinality of set A is written as (A).
Lets us answer the questions posted in the opening activity.
- How many sets are there? answer: there is the set if head covers (hats), the set of trees, the set of even numbers, and the set of polyhedra. But there are also set of round objects and set of pointy objects
- Does each belong to a set? answer: yes
- Is there an object that belongs to more than one set? which ones? answer: All the hats belong to the set of round objects, the pine trees, and two polyhedra belong to the set of round objects.
Do the following exercises
1. Give 3 examples of well-defined sets.
a. set of the whole number
b. set B
c. set N
2. Name two subsets of the set of whole numbers using both listing and rule method.
1. E = {0,2,4,6,8,10,...}
2.O = {x|x odd numbers}
3. Let B = {1,3,5,7,9}. List all the possible subsets of B.
1.{1,3}
2.{1,5}
3.{1,7}
4.{1,9}
5.{1,3,5}
6.{1,3,7}
156
B Venn Diagrams
Sets and subsets may be represented by using Venn diagrams. These are diagrams that make use of geometric shapes to show relationships between sets. Let the universal set U be all the elements in the set A, B, C, and D.
Each shape represents a set. Note that although there are no elements inside each shape, we can surmise how the sets are related to each other. Notice that set B is inside set A. This indicates that all elements in B are in contained in A. Same with set C. D, however, is separated fromA, B, and C. What does it mean?
Exercises
Draw a Venn Diagram to show the relationships between the following pairs or group sets
1. E = {2,4,8,16,32}
F = {2,32}
2. V is the set of all numbers
W = {5,15,25,35,45,55,...}
3. R = {x|x is a factor of 24}
S = {}
T = {7,9,11}
Summary
In this lesson, you learned about sets, subsets, the universe set, the null set and the cardinality of a set. You also learned to use the Venn diagram to show relationships between sets.
Lesson 2.1 Union and Intersection of sets
Prerequisite Concepts: Whole Numbers, Defenition of sets, Venn diagrams
About the lesson:
After learning some introductory concepts about sets, a lesson on set operations follows. The student will learn how to combine set (union) and how to determine elements common in 2 to 3 sets (intersection).
OBJECTIVES:
In this lesson, you are expected to:
Lesson Proper
1. Activities
ANSWER THE FOLLOWING QUESTIONS:
1.Which of the following shows the union of set A and B?
Sets and subsets may be represented by using Venn diagrams. These are diagrams that make use of geometric shapes to show relationships between sets. Let the universal set U be all the elements in the set A, B, C, and D.
Each shape represents a set. Note that although there are no elements inside each shape, we can surmise how the sets are related to each other. Notice that set B is inside set A. This indicates that all elements in B are in contained in A. Same with set C. D, however, is separated fromA, B, and C. What does it mean?
Exercises
Draw a Venn Diagram to show the relationships between the following pairs or group sets
1. E = {2,4,8,16,32}
F = {2,32}
W = {5,15,25,35,45,55,...}
3. R = {x|x is a factor of 24}
S = {}
T = {7,9,11}
In this lesson, you learned about sets, subsets, the universe set, the null set and the cardinality of a set. You also learned to use the Venn diagram to show relationships between sets.
Lesson 2.1 Union and Intersection of sets
Prerequisite Concepts: Whole Numbers, Defenition of sets, Venn diagrams
About the lesson:
After learning some introductory concepts about sets, a lesson on set operations follows. The student will learn how to combine set (union) and how to determine elements common in 2 to 3 sets (intersection).
OBJECTIVES:
In this lesson, you are expected to:
- Describe and define: (a) union of sets; (b) intersection of sets
- Perform the set operations: (a) union of sets; (b) intersection of sets
- Use Venn Diagram to represent the union and intersection of sets.
Lesson Proper
1. Activities
1.Which of the following shows the union of set A and B?
2. Which of the following show the intersection of set A and B?
how many elements are in intersection A and B?
here's another activity:
let
V={x|x ∈ I, 1 ≤ x ≤ 4}
W={x|x ∈ I, -2 ≤ x ≤ 2}
What elements may be found in intersection V and W? How many are there?
What elements may be found in union V and W? How many are there?
Do you remember how to use Venn diagrams? Based on the diagram below, (1) determine the elements that belong to both A and B
(2) determine the elements that belong to A or B or both. How many are there?
Important terms/symbols to remember the following are terms that you must remember from this point now on.
- Let A and B be sets .then union the union of A and B, Denoted by A∪B, is the set that contains those elements that belong to A, B or both. an element x belongs to the union of sets A and B. If and only if x belongs to A or B or both, this tells us that: A∪B={x|x ∈ A or x ∈ B} Using the Venn diagram, all of the set A and of B are shaded to show A∪B.
Let A and B be sets. The intersection of the sets A and B, denoted by A∩B is the set containing the elements that belong to both A and B. An element x belongs to the intersection of set A and B. If only x belongs to A and B. This tells us that: A∩B={x|x ∈ A or x ∈ B} Using the Venn diagram, the set A∩B consists of the shared regions of A and B.
Sets whose intersection is an empty set is called disjoint sets.
- The cardinality of the union of two sets is given by the following n(A∪B) = n(A) + n(B) - n(A∩B).
Let us answer the questions posted in the opening activity.
- Which of the following shows the union A and B? set 2. This is because it contains all the elements that belong to A or B or both. there are 8 elements.
- Which of the following shows the intersection of set A and B? set 3. this is because it contains all elements that are both in A and B. There are 3 elements.
V={2,4,6,8}
W={0,1,4}
Therefore, V∩W={4} has 1 and V∪W={0,1,2,3,4,6,8} has 6 elements. Note that the element {4} was only counted once.
On the Venn diagram:(1) the set that contains elements that belong to both A and B consist of two elements {1,12}; (2) the set that contains elements that belong to A or B or both consist of 6 elements {1,10,12,20,25,36}
|||. Exercises'
1. Given sets A and B.
determine which of the following shows (a) A∪B; and (b) A∩B
Answers: A = set 4 B = set 2
Set A:
Students who play guitar
|
Set B:
Students who play piano
|
Ethan Molina
|
Mayumi Torres
|
Chris Clemente
|
Janis Reyes
|
Angelina Dominguez
|
Christine Clemente
|
Mayumi Torres
|
Ethan Molina
|
Joanna Cruz
|
Nathan Santos
|
Set 1
|
Set 2
|
Set 3
|
Set 4
|
Ethan Molina
|
Mayumi Torres
|
Mayumi Torres
|
Ethan Molina
|
Chris Clemente
|
Ethan Molina
|
Chris Clemente
|
Chris Clemente
|
Angelina Dominguez
|
Chris Clemente
|
Janis Reyes
|
Janis Reyes
|
Mayumi Torres
|
Ethan Molina
|
Mayumi Torres
|
|
Joana Cruz
|
Nathan Santos
|
||
Joana Cruz
|
|||
Angelina Dominguez
|
2. Do the following exercises. Write your answers on the spaces provided
A={0,1,2,3,4} B={0,2,4,6,8} C={1,3,5,7,9}
Given the sets above, determine the elements and cardinal of:
a. A∪B={0,1,2,3,4,5,7,9} = 8 elements
b. A∪C={0,1,2,3,4,6,8} = 7 elements
c. A∪B∪C={0,1,2,3,4,5,6,7,8,9} = 10 elements
d. A∩B={0,2,4} = 3 elements
e. A∩C={0,1,3} = 3 elements
f. A∩B∩C={} = 0 elements
g. (A∩B)∪C={0,1,2,3,4,5,7,9} = 8 elements
A={0,1,2,3,4} B={0,2,4,6,8} C={1,3,5,7,9}
Given the sets above, determine the elements and cardinal of:
a. A∪B={0,1,2,3,4,5,7,9} = 8 elements
b. A∪C={0,1,2,3,4,6,8} = 7 elements
c. A∪B∪C={0,1,2,3,4,5,6,7,8,9} = 10 elements
d. A∩B={0,2,4} = 3 elements
e. A∩C={0,1,3} = 3 elements
f. A∩B∩C={} = 0 elements
g. (A∩B)∪C={0,1,2,3,4,5,7,9} = 8 elements
3. Let W = {x|0 < x < 3}, Y = {x|x > 2}, and Z = {x|0 ≤ x ≤ 4}
then determine (a) (W∩Y); (b) W∩Y∩Z.
answer:
(a) = {0,1,2,3,4,5,...}
(b) = {0,1,2,3,4,5,...}
Summary
In this lesson, you learned the definition of union and intersection of sets.You also how to use the Venn diagram to represent the union and intersection of sets. You also learned how to determine the elements that belong to the union and intersection of sets.
lesson 2.2: complement of a set
Prerequisite Concepts: sets, universal set, empty set, union and intersection of sets, cardinality of sets and Venn diagrams.
About the lesson:
The compliment of a set is an important concept. There will be times when one needs to consider the elements not found in a particular set.
A. You must know that this is when you need the compliment of a set.
Objective
In this lesson, you are expected to:
- Describe and define the compliment of a set.
- Find the compliment of the given set.
- Use Venn diagram to represent.
1. Problem
In a population of 8000, 2100 are freshmen, 2000 are sophomore, 2050 are Juniors and the remaining 1850 are either in their fourth or fifth year in their university. A student was selected from the 8000 and it is not a sophomore, how many possible choices are their?
answer: 6000
Discussion
Definition: The complement of a set A, written as A', is the set of all elements found in the universal set, U, that are not found in set A. The cardinality n (A') is given by
n(A) = n(U) - n(A)
Venn diagram
Examples:
1. Let U = {0,1,2,3,4,5,6,7,8,9}, and A = {0,2,4,6,8}. Then the A' are the elements of U that are not found in A therefore, A' = {1,3,5,7,9} and A = 5
2. Let U = {1,2,3,4,5}, A = {2,4} and B = {1,5}. Then
A' = {1,3,5}
B = {2,3,4}
A' ∪ B' = {1,2,3,4,5} = U
3. Let U = {1,2,3,4,5,6,7,8}, A = {1,2,3,4} and B = {3,4,7,8} then
A' = {5,6,7,8}
B' = {1,2,5,6}
A'∩B' = {5,6}
4. Let U = {1,3,5,7,9}, A = {5,7,9} and B = {1,5,7,9}
A∩B = {5,7,9}
(A∩B)' = {1,3}
5. Let U = {x| x is a whole number and x > 10}, then A = {x|x is a whole number and 0 ≤ x ≤ 10}.
The opening problem asks for how many there are for a student that is selected and known to be a non-sophomore. Let U be the set of all students and n(U) = 8000. Let A be the set of all Sophomores then n(A) = 2000. The set A consists of all students in U that are not sophomores and n(A') = n(U) - n(A) = 6000. There are 6000 possible choices for that selected student.
Activity
Shown in the table are names of students of a high school class by sets according to the definition of each set
A
Likes Singing
|
B
Likes Dancing
|
C
Likes Acting
|
D
Don’t like any
|
Jasper
Faith
Jacky
Miguel
Joel
|
Charmaine
Leby
Joel
Jezryl
|
Jacky
Jasper
Ben
Joel
|
Billy
Ethan
Camille
Tina
|
a.
U = {Jasper, Jacky, Miguel, Joel, Charmaine, Leby, Jezryl,
Ben, Billy, Ethan, Camille, Tina}
b.
A ∪ B’ = {Jasper, Faith,
Jacky, Miguel, Joel, Ben, Billy, Ethan, Camille, Tina}
c.
A’ ∪ C = {Jasper, Jacky,
Joel, Charmaine, Leby, Jezryl, Ben, Billy, Ethan, Camille, Tina}
d.
(B ∪ D)’ = {Jasper, Faith, Jacky,
Miguel, Ben}
e.
A’ ∩ B = {Leby, Charmaine, Jezryl}
f.
A’ ∩ D’ = {Leby, Charmaine, Jezryl, Ben}
g.
(B ∩ C)’ = {Jasper, Faith, Jacky, Miguel, Charmaine, Leby,
Jezryl, Ben, Billy, Ethan, Camille, Tina}
Exercises
1. True or False. If your answer is false give the correct answer.
let U = the set of months of the year
X = {March, May, June, July, October}
Y = {January, June, July}
Z = {September, October, November, December}
a. Z' = {January, February, March, April, May, June, July, August}
true
b. X' ∩ Y' = {June, July}
{October}
Exercises
1. True or False. If your answer is false give the correct answer.
let U = the set of months of the year
X = {March, May, June, July, October}
Y = {January, June, July}
Z = {September, October, November, December}
a. Z' = {January, February, March, April, May, June, July, August}
true
b. X' ∩ Y' = {June, July}
{October}
c. X' ∪ Z' = {January, February, March, April, May, June, July, August, September, November, December}
{January, February, April}
d. (Y∪Z)' = {February, March, April, May}
{true}
2. Place the elements in their respective sets in the diagram below based on the following elements assigned to each set
U = {a, b, c, d, e, f, g, h, i, j}
A = {a, b, c, d, e, g, j}
B = {a, b, d, e, h, i}
C = {a, b, c, f, i, j}
3. Draw a Venn diagram to show the relationships between sets U,X,Y and Z, given the following information
U, the universal set contains set X, set Y, and set Z
answer:
X∪Y∪Z = U
answer:
Z is a compliment of X
answer:
Y' includes some elements of X and Z
answer:
Summary
In this lesson, you learned about the compliment of a given set. You learned how to describe and define the compliment of a set, and without relates, to the universal set, U and the given set
