Module 1: NUMBERS AND NUMBER SENSE
Lesson 1: SETS
Learning Competency 3.3: Use Venn diagram to represent sets, subsets and set
operations.
I – OBJECTIVES
a. Describe and define the:
i. complement of a set.
ii. difference between sets
b. Find the complement of a set and the difference between sets.
c. Use Venn diagrams to represent complement of a set and set difference.
d. Value accumulated knowledge as means of new understanding.
II – SUBJECT MATTER
Topic: SETS
Subtopic: Set Operations and Venn diagrams.
References: TG-G7 pp.11-13, Grade 7 Mathematics Patterns and Practicalities by
Gladys S. Nivera, pp. 18-21
III – PROCEDURE
A. Preliminaries:
Motivational Activity:
The municipality of Maragondon held an annual dance contest for Dagundong
Festival. Students from different schools and chosen municipal employees
participated in the said contest. Suppose students of Bucal National School, students
of Maragondon National and students of PUP were held as winners. Who lost in the
contest?
B. Lesson Proper
1. Teaching/Modeling
Exploratory Activity
Grade 7 And Grade 8 teachers of Maragondon National High School has
spent their summer vacation in the islands of Calaguas and Caramoan.
Grade 7 teachers had their trip to Calaguas , they camped on the white sandy
beach, swam in its clear blue waters and hiked onto the peak of the hill to see
the entire barrio below. They visited nearby islets, too.
Grade 8 teachers went to an island hopping tour in Caramoan and discovered
its mysterious bangus in a sea pond, they had fun lounging on its white
sandy beaches. They also had an awesome time snorkeling under its blue
waters and seeing different kinds of corals and fishes.
But unlike in Boracay, both islands do not offer other water activities like
banana boat ride, paragliding and wind surfing.
2. Analysis
2.. Analysis
a. What are the activities the teachers did in Calaguas? In Caramoan?
b. What characteristics do these two islands have?
c. List the activities that they can do both on these islands.
d. List all the activities that either if these islands had to offer or both.
e. What activities they couldn’t do there?
3. Guided Practice
1. Let U (the universal set) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (a subset of a positive
integers)
A = {2, 4 ,6, 8}
B = {1, 2, 3, 4, 5}
Find the following:
1 (𝐴 ∪ 𝐵 ) ′
2. (𝐴 ∩ 𝐵)′
3. 𝐴′
4. 𝐵′
2: Given set A = {b, d, e, g, a, f, c}
set B = { k, h, u, a, f, c}
Draw Venn diagram and find:
1. A – B
2. B – A
4. Independent Practice
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 4, 6, 8, 10}
B = {1, 3, 5, 7, 8, 9}
Find:
(a) A'
(b) B'
(c) A' ∪ B'
(d) A' ∩ B'
(e) (A ∪ B)'
Also show (A ∪ B)' = A' ∩ B'.
5. Generalization
If we cut out set A from the picture on the left, the
remaining region is U, the universal set is labeled
𝐴′ and is called the complement of a set.
The complement of set A is all of the elements (in
the universe) that are NOT in set A.
NOTE*: The complement of a set can be represented with
several differing notations.
The complement of set A can be written as
𝑨′ or 𝑨c or Ā or Ã
Difference between Sets
Let A and B be sets. The difference of A and B, denoted by A - B, is the set
containing those elements that are in A but not in B. The difference of A and B
is also called the complement of B with respect to A.
The difference of B and A, denoted by B – A, is the set containing those
elements that are in B but not in A. It is called the complement of A with
respect to B.
Example:
If A = { 1, 2, 3} and B ={ 3, 4, 5},
Then
A - B = { 1, 2 }
B – A = {4 , 5}
A–A={}
B–B={}
6. Application
If A = {a, b, c, d} B = {b, c, d, e}
C = {c, d, e, f} D = {d, e, f, g},
Find:
(a) A - B (e) B - A
(b) B - C (f) C - B
(c) C - D (g) D - C
(d) D - A (h) A - D
7. Assessment
The table shows the different school activities that the following students had
participated last year.
A B C D
Math Camp Masquerade Party Educational Trip Not attended any
Ruth Rhojan Kiel Zyra
Shiela Katrina Jet John Carlo
Marielle Camil Ruth Timothy
Rhojan Mae Rhojan
Camil
Draw Venn diagram. List the elements and cardinality of the following sets.
1. 𝐴 ∪ 𝐵 ∪ 𝐶
2. (𝐴 ∪ 𝐵)′
3. (𝐴 ∪ 𝐶 )′
4. (𝐵 ∪ 𝐶)′
5. (𝐴 ∩ 𝐵 ∩ 𝐶)′
6. 𝐴 − 𝐵
7. 𝐵 − 𝐶
8. 𝐶 − 𝐴
9 (𝐴 ∪ 𝐵) − 𝐶.
10. 𝐴 ∪ (𝐵 − 𝐶)
IV. ASSIGNMENT
From the given Venn diagram , find the following:
1. (C ∩ E)’
2. E
3. C’ ∩ E
4. (E – C)’
5. (C ∪ E)’
http://www.regentsprep.org/regents/math/algebra/ap2/lvenn.htm
http://www.basic-mathematics.com/difference-of-sets.html
ANSWER KEY:
Analysis:
Illustrative example: Using Venn diagram we can easily see the union,
intersection of sets and set complement.
Let U be the universal set
U = {camping, swimming, hiking, island hopping, snorkeling, visiting
pond, paragliding, windsurfing, banana boat riding}
Let A be set of Calaguas activities
A = {camping,swimming,hiking,island hopping}
Let B be set of Caramoan activities
B = {swimming,island hopping, snorkeling,visiting pond}
The union of set A and set B written as 𝑨 ∪ 𝑩 are all activities that you can
do either in Calaguas or in Caramoan or both islands.
𝐴 ∪ 𝐵 = {𝑐𝑎𝑚𝑝𝑖𝑛𝑔, 𝑠𝑤𝑖𝑚𝑚𝑖𝑛𝑔, ℎ𝑖𝑘𝑖𝑛𝑔, 𝑖𝑠𝑙𝑎𝑛𝑑 ℎ𝑜𝑝𝑝𝑖𝑛𝑔, 𝑠𝑛𝑜𝑟𝑘𝑒𝑙𝑖𝑛𝑔,
𝑣𝑖𝑠𝑖𝑡𝑖𝑛𝑔 𝑝𝑜𝑛𝑑}
The intersection of set A and set B written as 𝑨 ∩ 𝑩 are the activities that
both islands have in common.
𝐴 ∩ 𝐵 = {𝑠𝑤𝑖𝑚𝑚𝑖𝑛𝑔, 𝑖𝑠𝑙𝑎𝑛𝑑 ℎ𝑜𝑝𝑝𝑖𝑛𝑔}
c
The complement of set A written as 𝑨′ or 𝑨 are activities that you can not
do in Calaguas
𝑠𝑛𝑜𝑟𝑘𝑒𝑙𝑖𝑛𝑔, 𝑣𝑖𝑠𝑖𝑡𝑖𝑛𝑔 𝑝𝑜𝑛𝑑, 𝑝𝑎𝑟𝑎𝑔𝑙𝑖𝑑𝑖𝑛𝑔, 𝑤𝑖𝑛𝑑𝑠𝑢𝑟𝑓𝑖𝑛𝑔,
𝐴′ = { }
𝑏𝑎𝑛𝑎𝑛𝑎 𝑏𝑜𝑎𝑡 𝑟𝑖𝑑𝑖𝑛𝑔
The complement of set B written as 𝑩′ or 𝑩c are activites that you can not
do in Caramoan
𝐵′ = {𝑐𝑎𝑚𝑝𝑖𝑛𝑔, ℎ𝑖𝑘𝑖𝑛𝑔, 𝑝𝑎𝑟𝑎𝑔𝑙𝑖𝑑𝑖𝑛𝑔, 𝑤𝑖𝑛𝑑𝑠𝑢𝑟𝑓𝑖𝑛𝑔, 𝑏𝑎𝑛𝑎𝑛𝑎 𝑏𝑜𝑎𝑡 𝑟𝑖𝑑𝑖𝑛𝑔}
The complement of the union of sets A and B written as (𝑨 ∪ 𝑩)′ are
activities that you can not do on both islands.
(𝐴 ∪ 𝐵)′ = {𝑝𝑎𝑟𝑎𝑔𝑙𝑖𝑑𝑖𝑛𝑔, 𝑤𝑖𝑛𝑑𝑠𝑢𝑟𝑓𝑖𝑛𝑔, 𝑏𝑎𝑛𝑎𝑛𝑎 𝑏𝑜𝑎𝑡 𝑟𝑖𝑑𝑖𝑛𝑔}
The complement of the intersection of sets A and B written as (𝐴 ∩ 𝐵)′ are
beach activities except swimming and island hopping.
(𝐴 ∩ 𝐵)′ = {𝑐𝑎𝑚𝑝𝑖𝑛𝑔, ℎ𝑖𝑘𝑖𝑛𝑔, 𝑠𝑛𝑜𝑟𝑘𝑒𝑙𝑖𝑛𝑔, 𝑣𝑖𝑠𝑖𝑡𝑖𝑛𝑔 𝑝𝑜𝑛𝑑, 𝑝𝑎𝑟𝑎𝑔𝑙𝑖𝑑𝑖𝑛𝑔,
𝑤𝑖𝑛𝑑𝑠𝑢𝑟𝑓𝑖𝑛𝑔, 𝑏𝑎𝑛𝑎𝑛𝑎 𝑏𝑜𝑎𝑡 𝑟𝑖𝑑𝑖𝑛𝑔}
Activities you can do only in Calaguas is 𝐴 − 𝐵 = {𝑐𝑎𝑚𝑝𝑖𝑛𝑔, ℎ𝑖𝑘𝑖𝑛𝑔}
Activities you can do only in Caramoan is 𝐵 − 𝐴 = {𝑠𝑛𝑜𝑟𝑘𝑒𝑙𝑖𝑛𝑔, 𝑣𝑖𝑠𝑖𝑡𝑖𝑛𝑔 𝑝𝑜𝑛𝑑}
GUIDED PRACTICE:
1.
𝑨 ∪ 𝑩 = {1,2,3,4,5,6,8} Union – ALL elements in both sets
𝑨 ∩ 𝑩 = {2,4} Intersection – elements where sets overlap or
common elements of both sets
𝑨′ = {1, 3, 5, 7, 9, 10} Complement – elements NOT in the set.
𝑩′ = {6, 7, 8, 9, 10}
Therefore:
(𝑨 ∪ 𝑩)′ = {7, 9, 10}
(𝑨 ∩ 𝑩)′ = {1, 3, 5, 6, 7, 8, 9, 10}
𝟐. the set difference of set B from set A is the set of all element in A, but not in B.
We can write A – B
Fig.1
Take a close look at the figure above. Elements in A only are b, d, e,g.
Therefore, A − B = { b, d, e, g}
Notice that although elements a, f, c are in A, we did not include them in
A − B because we must not take anything in set B.
Sometimes, instead of looking at a the Venn Diagrams, it may be easier
to write down the elements of both sets
Then, we show in bold the elements that are in A, but not in B
A = {b, d, e, g, a, f, c}
B = { k, h, u, a, f, c}
The set difference of set A from set B is the set of all elements in B,
but not in A.
Written as B – A
Fig. 2
Looking at Fig.2, the elements in B only are u, h, k.
Therefore, B – A = {k, h, u}.
From the list of elements below, we show in bold the elements in B, but
not in A.
A = {b, d, e, g, a, f, c}
B = { k, h, u, a, f, c}
INDEPENDENT PRACTICE:
1. A’ = {3,5,7,9}
2. B’ = {2,4,6,10}
3. 𝐴′ ∪ 𝐵′ = {2,3,4,6,7,9,10 }
4.𝐴′ ∩ 𝐵′ = { }
5. 𝐴 ∪ 𝐵 = {1,2,3,4,5,6,7,8,9,10} = 𝑈
(𝐴 ∪ 𝐵)′ = { } 𝑠𝑖𝑛𝑐𝑒 𝐴 ∪ 𝐵 = 𝑈 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 𝑛𝑜 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝐴 ∪ 𝐵.
Then (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′
APPLICATION
a. {a}
b. {b}
c. {c}
d. {e,f,g}
e. {e}
f. {f}
g. {g}
h. {a,b,c}
ASSESSMENT
1. {Shiela, Marielle, Ruth, Camil, Rhojan, Mae,
Katrina, Kiel, Jet, Zyra, Timothy, John Carlo
2.{ Kiel, Jet, Zyra, Timothy, John Carlo}
3. {Mae, Katrina, Zyra,Timothy, John Carlo}
4. {Shiela, Marielle, Timothy, Zyra, John Carlo}
5. {Shiela, Marielle, Ruth, Camil, Mae, Katrina, Zyra,
Timothy, John Carlo}
6. {Shiela, Marielle, Ruth}
7. {Camil, Mae, Katrina}
8. {Kiel, Jet}
9. {Shiela, Marielle, CAmil, Mae, Katrina}
10. {Shiela, Marielle, CAmil, Mae, Katrina}
ASSIGNMENT
1. {2,3,17, 5,6,20, 11,4}
2.{10,13,5,6,20}
3. {5,6,20}
4.{ 4, 11, 2,3, 17,10,13}
5. {11,4}
