Republic of the Philippines
Department of Education
Region III
Division of City of San Fernando (P)
MAGLIMAN INTEGRATED SCHOOL
DETAILED LESSON PLAN
IN MATHEMATICS 7
(Quarter 1 - Week 1 - Day 2)
I. Objectives
A. Content Standards
The learner demonstrates understanding of key concepts of sets and the real number system.
B. Performance Standards
The learner is able to formulate challenging situations involving sets and real numbers and solve
these in a variety of strategies.
C. Learning Competencies
The learner describes well-defined sets, subsets, universal sets, and the null set and cardinality
of sets. [M7NS-Ia-1]
D. Objectives
At the end of the lesson, you should be able to:
II. Content
UNIVERSAL AND SUBSETS, EQUAL AND EQUIVALENT SETS
Learning Resources
A. Reference
Second edition “Next Century Mathematics” by Fernando B. Orines [Revisor}, page 2 - 8
B. Other Learning Resources
Practical Mathematics, pp 5 - 14
III. Procedures
A. Reviewing previous lesson or presenting the new lesson
How’s your day?
Are you now ready to learn another lesson?
So let us start by answering the activity 1.
Activity 1
Given:
Let A be the set of the letters of the English alphabet.
Let B be the set of vowels.
Let C be the set of consonants.
Let D be the set of the letters of the word DARES.
Let E be the set of the letters of the word READS.
Your Task!
1. List the elements of set A
2. List the elements of set B
3. List the elements of set C
4. Express set A in set-builder notation
5. Express set B in set-builder notation
6. Express set C in set-builder notation
7. List the elements of sets D and E
8. Determine the cardinality of each set
Now, check your work by turning to page 5 for the key to correction. If your score is at least 6 out of 8,
you may now proceed to next part of the discussion. If not, I am sorry but you have to go back to the
activity and try all over again.
B. Establishing a purpose for the lesson
Base from the activity 1 in Section A part, answer the following questions:
a. Which set contains all the elements of all the other sets?
b. Which set/s contain/s elements that is/are found in set A?
c. Which sets have the same number of elements?
d. Which sets have identical/exactly the same elements?
C. Presenting examples/instances of the new lesson
Let’s try this!
Let U = {Presidents of the Philippines}
P = {Corazon Aquino, Ramon Magsaysay Jr., Diosdado Macapagal, Rodrigo Duterte}
N = {Jose Rizal, Andres Bonifacio, Ferdinand Marcos}
S = {Grace Poe, Manny Villar, Manuel Roxas, Jejomar Binay}
O = {Antonio Trillanes, Ferdinand Marcos Jr., Leni Robredo}
Q = {Ferdinand E. Marcos, Fidel V. Ramos, Joseph E. Estrada}
Task: Examine the given sets.
Questions:
1. Which sets are subsets of set U?
2. Give all the possible subsets of Q.
3. Which sets are equivalent?
4. Which sets are equal? Explain your answer.
Check your work by turning to page 5 for the key to correction. If you do not get the correct answer, you
have to go back to the activity and try all over again.
D. Discussing new concepts and practicing new skills #1
Concept
Definition of a Set
Before we get into the definition of an equivalent set, we need to first know what a set is.
A set is a collection of elements that are usually related. They are indicated with brackets: { }.
We can have a set containing numbers, words, or even pictures.
Here are some examples of sets:
{January, March, May, November}
{1, 2, 3, 4, 5, 6}
When a set continues on for infinity, the last element in the set is followed by three dots
known as an ellipsis, which indicates that the numbers continue.
An example is shown here: {1, 2, 3, 4, 5, 6. . . }.
Universal set
Basic operation are also done on a pair of sets to obtain a new set. In basic set of
operations, the concept of universal is important. In any set operation, all the sets under
investigation will likely be subsets of a universal set which is usually denoted by capital
letter U.
For instance,
- the set of fractions is a subset of the universal set of numbers
- the Philippine population is a subset of the universal set of population
- a line in a plane is a subset of the universal set of all points in the plane
Subsets
If all the elements of set A are also elements of set B, then A is a subset of B. This
relation between sets A and B can be written in symbol as
A⊆B
which is read as “ A is a subset of B” or “A is contained B”
Consider the following sets.
L = {1, 2, 3, 4, 5} M = {5, 4, 3, 2, 1} N = {1, 4, 3} O = {2, 4, 6} P={ }
Set M is a subset of L because every element of M is also an element of L. Thus,
M⊆L
Note that L is also a subset of M.
L⊆M
This means that L and M are equal sets.
Equality of Sets
Two sets A and B are equal, that is
Set N is a subset of L because all elements of N are also elements of L. But N is not
equal to L. In symbols,
N≠L
Specially, N is a proper subset of L. In symbols, N ⊂ L
Proper Subset A ⊂ B
Set A is a proper subset of set B if
if and only if A ⊆ B and A ≠ B
In the given illustrate example, set O is not a subset of L because there is an element 6 of set O
which is not an element of L. In symbols,
O⊄L
Set P is a null set, and is considered a subset of any set. Remember that the null set is a subset
of every set.
Example: Write all the subsets of each set and identify all the proper subsets.
e. Q = {x, y} b. R = { ▲, ,,}
Solution:
a. The subsets of Q are {x}, {y}, {x, y}, and { }. The proper subsets of Q are {x} and
{y}.
b. The subsets of R are {▲},{ }, {}, { ▲, }, { ▲,,},{ , }, { ▲, ,,}, { }
The proper subsets of R are {▲},{ }, {}, { ▲, }, { ▲,,} and { ,}
E. Discussing new concepts and practicing new skills #2
Did you know!
Maybe you’re confused with the difference between equal and equivalent sets, kindly read the
concept below.
Equal and Equivalent Sets
When we have two sets that have the exact same elements, we call them equal sets. It
does not matter what order the elements are in. It just matters that the same elements are in
each set.
Here are some examples of equal sets:
{1, 3, 5, 7} and {7, 5, 3, 1}
{January, March, May, November} and {May, March, January, November}
An equivalent set is simply a set with an equal number of elements. The sets do not
have the same exact elements, just the same number of elements.
Let's take a look at some examples:
Example 1
Set A: {A, B, C, D, E}
Set B: {January, February, March, April, May}
Even though Sets A and B have completely different elements (Set A comprises letters,
and Set B comprises months of the year), they have the same amount of elements, which is
five. Set A contains five letters and Set B contains five months. That makes them equivalent
sets!
Example 2
Set C: {Sweater, Mittens, Scarf, Jacket}
Set D: {Apples, Bananas, Peaches, Grapes}
Set C and Set D both comprise word elements in completely different categories (Set C
comprises articles of clothing you would wear when cold, and Set D comprises types of fruit),
but they both have the same amount of elements, which is four. That makes them equivalent
sets!
FYI…
Here are the set symbols for today’s lesson
In example A = {1, 2, 3, 4} and B = {3, 4, 5}
Symbol Meaning Example
{} Set: a collection of {1, 2, 3, 4}
F.
elements
A⊆B Subset: A has some (or {3, 4, 5} ⊆ B
all) elements of B
A⊂B Proper Subset: A has {3, 5} ⊂ B
some elements of B
A⊄B Not a subset: A is not a {1, 6} ⊄ B
subset of B
A B Superset: A has some {1, 2, 3} {1, 2, 3}
element as B, or more
A B Proper Superset: A has {1, 2, 3, 4} {1, 2, 3}
B’s elements and more
Developing mastery
Refer to the given sets and tell whether the statement is true or false.
U = {Tourist destinations in Bicol}
A = {Mayon Skyline, Kawa-kawa Natural Park, Bambusetum, Caramoan Islands,
Vanishing Island}
B = {Camalig, Guinobatan, Ligao, Mt. Isarog}
C = {3, 6, 9, 12, 15}
D = {first five non-zero multiples of 3}
1. A ⊂ 𝑈
2. A ⊆ 𝐵
3. B ⊂ 𝑈
4. C and D are equal sets.
5. n(U) = n(D)
6. ∅ ⊂ B
7. A is equivalent to C
8. C ⊄ {all multiples of 3}
9. Boracay ∈ U
10. D ⊆ C
G. Finding practical applications of concepts and skills in daily living
1. Give all the possible subsets of your family.
2. Is your section a subset of the school community?
H. Making generalizations and abstractions about the lesson
As for our summary of today’s lesson:
a. What is a universal set?
In any set operation, all the sets under investigation will likely be subsets of a universal
set which is usually denoted by capital letter U.
For instance,
- the set of fractions is a subset of the universal set of numbers
- the Philippine population is a subset of the universal set of population
- a line in a plane is a subset of the universal set of all points in the plane
b. When is a set a subset of a given set?
If all the elements of set A are also elements of set B, then A is a subset of B. This
relationbetween sets A and B can be written in symbol as
A⊆B
which is read as “ A is a subset of B” or “A is contained B”
c. When are two sets equal? Are equal sets equivalent? Why or why not?
When we have two sets that have the exact same elements. No, because equivalent is
simply a set with an equal number of elements. The sets do not have the same exact
elements, just the same number of elements.
I. Evaluating learning
Do this!
Insert the correct symbol ⊂,⊃,⊆,⊇,⊄,⊅,⊈,⊉ or = to make the relationship true.
Given: U = {positive integers less than 15}
A = {factors of 12}
B = {multiples of 5 less than 15}
C = {5,10} D = {1,2,3,4,5,6} E = { }
Problems:
1. A __ U
2. E __ D
3. C __ U
4. B __ C
5. n(A) __ n(D)
J. Additional activities for application or remediation
Given: U = {1, 2, 3, 4, 5}
Create the following:
1. a subset
2. 2 equal sets 3.
2 equivalent sets
If you need more help, you may reach me at cp # 09487830826 / 09556459447 or send me a private
message thru my facebook account jhoygadil@yahoo.com .
Prepared by:
MARY – ANNE JOY G. VIRAY
Teacher I
Key to Correction:
Section A.Your Task
1. Set A = {a, b, c, d, e,…,z}
2. Set B = {a, e, i, o, u}
3. Set C = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, x, z}
4. A = {x x is the set of the letters of English alphabet}
5. B = {x x is the set of vowels}
6. C = {x x is the set of consonants}
7. Set D = {D, A, R, E, S}
Set E = {R, E, A, D, S}
8. n(A) = 26, n(B)=5, n(C)= 21, n(D)= 5, n(E)=5
Section B.
a. Set A, Set D, and Set E
b. Set B, Set C, Set D and Set E
c. Set B, Set D and Set E
d. Set D and Set E
Section C. Let’s try this!
1. Set P and Set Q
2. ⊂ Q (null set is a subset of every set)
3. Set N, Set O and Set Q
4. None, because there is no set that have the same elements.
Section F.
1. True 6. True
2. False 7. True
3. False 8. False
4. True 9. False
5. True 10. True
Section G.
Answers may vary.
