Week 2 Math 7 Module

7 MATHEMATICS First Quarter – Module 2 Operations on Sets

What I Need to Know

This module is designed and written with you in mind. It is here to help you master Basic Set Operations. The scope of this module permits it to be used in many different ways. The language recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. This module is divided into two lessons, namely: 

Lesson 1 – Union and Intersection of Sets



Lesson 2 – Complement and Difference of Sets.

Based on the competency, this module will help you illustrate union, intersection, and difference of sets (M7NS-Ia-2). Specifically, after going through this module, you are expected to: 1. define and describe the different set operations; 2. perform set operations; 3. represent the set operations using the Venn Diagram. This module is self – instructional and allows you to learn in your own space and pace. So, relax and enjoy it!

1

MULTIPLE CHOICE Directions: Read the questions carefully. Choose the letter of your answer and write it in your Mathematics notebook. 1. What is the complement of U? A. U B. ∅ C. {0} D. {1} 2.

What is the meaning of the phrase “the intersection of P and Q”? A. The set of elements in the universe that is not in P. B. The set of elements in the universe that is not in Q. C. The set of elements is common to both P and Q. D. The set of elements that is in P or Q or both P and Q.

3.

If X = {Asia, Africa, North America, South America, Antarctica, Europe, Australia}, and Y = {Atlantic, Pacific, Arctic, Indian, Antarctic}, then which of the following could be the universal set? A. U = {oceans} C. U = {world} B. U = {countries} D. U = {planets}

4.

What is the meaning of the phrase “the union of P and Q"? A. The set of elements in the universe that is not in P. B. The set of elements in the universe that is not in Q. C. The set of elements is common to both P and Q. D. The set of elements that is in P or Q or both P and Q.

5.

Given P = {apple, orange, banana, mango} and Q = {orange, mango, watermelon}, what is P ∪ Q? A. P ∪ Q = ∅ B. P ∪ Q = {apple, orange, banana} C. P ∪ Q = {apple, orange, banana, mango} D. P ∪ Q = {apple, orange, banana, mango, watermelon } What is the complement of ∅ ? A. U B. ∅ C. {0}

6.

D. {1}

7.

Given X = {1, 4, 16} and Y = {1, 4, 9, 16, 25, 36}, what is X ∪ Y? A. X ∪ Y = ∅ C. X ∪ Y = {1, 4, 9, 16, 25} B. X ∪ Y = {1, 4, 16} D. X ∪ Y = {1, 4, 9, 16, 25, 36}

8.

Based on the given sets below, which statement is CORRECT? A = {1, 3, 6, 8, 9, 12, 15} and B = {6, 9, 12}? A. B ⊈ A C. A ∪ B = A B. A ∩ B = ∅ D. B is the complement of A. Given A = {even whole numbers} and B = {prime numbers }, what is A ∩ B? A. A ∩ B = ∅ C. A ∩ B = {0, 2} B. A ∩ B = {2} D. A ∩ B = {0, 2, 4}

9.

10. If M = {x I x is a number greater than 7} and N = {y I y is an even number less than 15}, what is M ∩ N? A. M ∩ N = ∅ C. M ∩ N = {8, 10, 12} B. M ∩ N = {8, 10} D. M ∩ N = {8, 10, 12, 14} 11. If U = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and A = { 12, 13, 14 }, what is A’? A. A’ = {12, 13, 14, 20} B. A’ = {11, 15, 16, 17, 18, 19} C. A’ = {11, 15, 16, 17, 18, 19, 20} D. A’ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} 12. Given U = {single digits} and B = {0, 1, 4, 5, 6, 7, 8}, what is B’?

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A. B’ = ∅ B. B’ = {2}

C. B’ = {2, 3} D. B’ = {2, 3, 9}

13. If set A = {3, 4, 5, 6} and set B = {2, 4, 6, 8 }, what is A – B ? A. A – B = ∅ C. A – B = {3, 5} B. A – B = {3} D. A – B = {4, 6} 14. If set A = {3, 4, 5, 6} and set B = {2, 4, 6, 8}, what is B – A? A. B – A = ∅ C. B – A = {3, 5} B. B – A = {2} D. B – A = {2, 8} 15. What does the shaded region of the Venn Diagram given below represent?

A. A ∪ B B. A ∩ B

Lesson 1

C. (A ∪ B)’ D. A – B

Union and Intersection of Sets

What’s In

Before we proceed to our lesson, let us see if we still remember our previous lesson by answering the given activity below. Activity 1: Arrange Me! Let’s Find Out: Terms Involved in Sets Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. Arrange the jumbled letters to get the correct answer. b. Write the answer in your Mathematics notebook. The first one is done for you! 1. It is a well – defined collection of distinct objects.

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E T S

SET

_____________________

2. It is the set of all possible elements of any set. L U A N S I R V E E T S 3. A set with no element. U L N L

E T S

4. Two sets that contain the same number of elements. T E N Q L U A I V E

T S E

5. It is a diagram that makes use of geometric shapes to show relationships between sets. N N V E

M D A I A G R

6. These are sets with a defined number of elements. E F T I I N 7. This is a method describing a set by listing each element of the set inside the symbol { }. R R O E T S 8. Two sets that contain the same elements. L E A Q U

T S E

9. These are sets having no element in common. T D N I I O S J 10. It is a method of describing a set in words. L V A E B R

Observe the given figures below. Figure 1

A

B

C

Figure 2

4

D

E

F

Notice that in Figure 1, the objects in Box C are the objects from Box A and B. If you combine the objects from Box A and B, the result is Box C. But take note, in Box C, there is no repetition of objects. On the other hand, Figure 2 illustrates that Box F is just a result if you get the common object from Box D and E. In Sets, combining the elements without repetition is called the Union of Sets while getting the common element is called Intersection of Sets. These are called Basic Set Operations.

What is It In arithmetic, we have Four Basic Operations such as addition, subtraction, multiplication, and division of numbers. In sets, we have also the Four Basic Set Operation. In this lesson, we will only discuss the first two operations namely: Union and Intersection of Sets. The last two operations will be discussed in the next lesson. Basic Set Operations Symbol

Meaning

∪

Union of Sets

1. Find the union of

 The set of

A = { 2, 3, 4} and

A∪B

Venn Diagram

Examples

elements

B = { 3, 4, 5}.

that belongs

Solution:

to set A or set

B

A ∪ B = { 2, 3, 4,

or

5}

read as A union B 5

both. A ∪ B is shaded

List all the elements in set A and all of the elements in set B. If an element is in both sets, we list it only once.

Reminder! To make it uniform if the elements of the given set are numbers then arrange your final answer in increasing order. Basic Set Operations Symbol

Meaning

∩

Intersection of

 of

intersection of

The set

A = { 2, 3, 4} and

elements

B = { 3, 4, 5}.

belongs

to

Solution:

both A and

A ∩ B = { 3, 4 }

B.  the

Examples 1. Find the

Sets

A∩B

read as An intersection B

Venn Diagram

Set

of

common

A ∩ B is shaded

elements in A and B.

The common elements of Sets A and B are 3 and 4.

2. Given: A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6}, and C = {1, 3, 5, …}

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Find: a. A ∩ C b. B ∩ C

A ∩ B ∩Cis shaded

Solution: a. A ∩ C = {1, 3, 5} b. B ∩ C = { } or ∅

No common element in sets B and C.

You did great! Now, we .have more examples. More Examples Given:

X = {2, 4, 6, 8, 10, 12},

Y = {3, 6, 9, 12, 15}

and Z = {1, 4, 7, 10, 13, 16}. Find: 1. X ∪ Y

3. X ∪ Z ∩ Y

2. Y ∩ Z

4. (Y ∩ X) ∪ Z

Solution: 1.

X∪Y

= {2, 4, 6, 8, 10, 12 } ∪ { 3, 6, 9, 12, 15 }

X

∪

Y First, list the given elements of Sets X and Y.

X ∪ Y = {2, 3, 4, 6, 8, 9, 10, 12, 15} Finally, get the union of X and Y. This means that combine the elements. If an element is in both sets, we list it only once.

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2.

Y ∩ Z = {3, 6, 9, 12, 15} ∩ {1, 4, 7, 10, 13, 16}

∩

Y

Z

First, list the given elements of Sets Y and Z.

Y∩Z={}

3.

Finally, get the intersection of Y and Z. Note that the intersection is the "common element". Since there is no common element, so the answer is a null or empty set.

X ∪ Z ∩ Y = {2, 4, 6, 8, 10, 12} ∪ {1, 4, 7, 10, 13, 16} ∩ {3, 6, 9, 12, 15}

X

∪

Z

∩

Y

First, list the given elements of sets X, Z, and Y.

X∪Z∩Y

= {1, 2, 4, 6, 7, 8, 10, 12, 13, 16}∩ {3, 6, 9, 12, 15}

Second, solve the union of sets X and Z. Here is the answer to X ∪ Z.

X∪Z∩Y

= { 6, 12 } Finally, solve the intersection of X ∪ Z and Y. Then, here is the final answer.

4.

( Y ∩ X ) ∪ Z = ({ 3, 6, 9, 12, 15 } ∩ { 2, 4, 6, 8, 10, 12 } ) ∪ { 1, 4, 7, 10, 13, 16 } 8

List the given elements of Sets Y, X, and Z.

(

Y

∩

) ∪

X

Z

First, list the given elements of sets Y, X, and Z.

( Y ∩ X ) ∪ Z = { 6, 12 } ∪ { 1, 4, 7, 10, 13, 16 } Second, solve the operation inside the parenthesis which is the intersection of Y and X. Here is the answer.

( Y ∩ X ) ∪ Z = { 1, 4, 6, 7, 10, 12, 13, 16 } Finally, get the union of the sets ( Y ∩ X ) and Z. Then, here is the final answer.

What’s More

Activity 2: Solve Me! Let’s Find Out: Union and Intersection of Sets Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. Solve the following based on the given below. b. Write the answer in your Mathematics notebook. The first one is done for you! Given: A = {2, 5}, B = {5, 7, 9}, C = {x I x is an odd number less than 9}, and D= {x I x is an even number less than 9}. 1. A ∩ C Solution:

A ∩ C = {2, 5} ∩ {1, 3, 5, 7} A ∩ C = {5} 9

2. (B ∪ D) ∩ C 3. A∩ B ∩ C 4. C ∪ (D ∩ A) 5. ( A ∪ C ) ∩ B

Excellent! Now you are ready for more Set Operations.

Lesson 2

Complement and Difference of Sets

What’s In

Before we proceed to our lesson, let us see if we still remember our previous lesson by answering the given exercise below. Activity 3: Find Me! Let’s Find Out: Hidden Words/ Terms Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. Find the given words below that are hidden in the grid. The words may be found vertically, horizontally, and diagonally. b. Write the answer in your Mathematics notebook. Y O J W

M H D G

K A N B

A L O Y

T E S P

J V G S

B R M T

K P X K

C X E G

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B L P Y

L I T B

D I L Z

K P S K

Z S X C

O P L X

U H N P M B I H N X R

L O E B N X N Y X D W

Z Y Z V S N T Y O K G

L E L N J X E P Z V V

M Q V O Y Q R V S A U

E A Z I K D S R T T C

J R P N I O E M F S N

J J A U T T C I P O B

D Y R T E C T U M P F

I M E P I T I M C Z S

O K E W O O O H X L K

X Q S E I C N F A C K

E L E M E N T S S L S

COMMON

ELEMENTS

INTERSECTION

OPERATION

VENN

UNION

J V M D L P U R L Z U

F V K A H Y I G H G P

SET

What’s New

Activity 4: Find Me Once Again! Let’s Find Out: The Vegetables in the Venn Diagram Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. Answer the following questions based on the Venn Diagram below.

1. What are the vegetables outside the pentagon? 2. How many vegetables are there inside the rectangle? 3. What are the vegetables that are inside the pentagon but not inside the circle?

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Great job! You did well in this activity. Get ready for more activities like this in our lesson.

What is It In the previous lesson, we have learned the union and intersection of sets. Now, we will discuss two more Set Operations which are the Complement of a set and Difference of two sets. Basic Set Operations Symbol A’ or Ac

Meaning

read as

Examples

A

Given:

complement

U = {1, 2, 3, 4, 5}

of a Set

and A = {1, 3, 5}.

 A complement

Venn Diagram

Set of

Find A’.

all elements

Solution:

in the

A’ = {2, 4}

universal set U that are not

A’ is shaded

in set A.

The complement of A is the set of elements in U but not in A. These elements are 2 and 4.

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Basic Set Operations Symbol A–B

Meaning Difference

Venn Diagram of

Given:

Sets

A = {3, 4, 5, 6} and

 The read as A minus B

Examples

set

B = {2, 4, 6, 8}

containing

Find:

elements of set A but

a. A – B

not in B.

b. B – A

 All

elements

of A except

A – B is shaded

Solution: a. A – B = {3, 5 }

the elements of B.

Note that the elements 4 and 6 are included in set B. Difference of sets A and B are set of elements in A but not in B.

b. B – A= {2, 8}

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Note that the elements 4 and 6 are included in set A. Difference of sets B and A are a set of elements in B

More Examples Given:

U = {1, 2, 3, 4, 5} A = {1, 3, 5} B = {3, 4, 5} and C = {2, 4} a.) A’ ∪ B’

Find:

b.) (A∩ C)’ c.) B – C’ Solution: a.)

A’ ∪ B’

=

{1, 3, 5}’

∪

A ’

∪

{3, 4, 5}’

B

’ First, list the elements of sets A and B.

A’ ∪ B’

=

{2, 4} ∪ { 1, 2} Second, solve A’ and B’. Here is the result.

A’ ∪ B’

=

{1, 2, 4} Finally, get the union of A’ and B’. Here is the final answer. 14

b.)

(A ∩ C )’

=

( {1, 3, 5}

∩

{ 2, 4} )’

(

∩

C

A

)’ First, list the elements of sets A and C. Second, solve the operation inside the parenthesis which is the intersection of A and C. Here is the answer.

(A ∩ C)’

=

({ } )’

(A ∩ C)’

=

{1 ,2, 3, 4, 5} Finally, get the complement. Note that the complement of an empty set is the universal set.

c.)

B – C’

=

{3, 4, 5}

–

{ 2 , 4 }’

B

–

C ‘ First, list the elements of Sets B and C.

B – C’

=

{3, 4, 5} – { 1, 3, 5 } Second, solve the complement of C. Here is the result.

B – C’

=

{4}

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Finally, solve the difference between B and C. Here is the

What’s More

Activity 5: Solve Me! Let’s Find Out: Complement and Difference of Sets Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. Solve the following based on the given below. b. Write the answer in your Mathematics notebook. The first one is done for you! Given:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 5}, B = {5, 7, 9}, C = {x I x is an odd number less than 9}, and D= {x I x is an even number less than 9}.

1. A – C Solution: A −¿ C

= {2, 5} −¿ {1, 3, 5, 7}

A −¿ C

= {2}

2. B – D’ 3. (A ∪ B)’ – C 4. D – (B ∩ A)

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What I Have Learned

Activity 6: Complete Me! Let’s Find Out: Terms Involved in Operations on Sets Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. Complete the following sentences. b. Write the answer in your Mathematics notebook. 1. The set of all elements in the universal set that is not in set A is called the _______________ of set A. 2. The set containing all the elements of set A or set B or both sets is called the _______________ of set A and set B. 3. The set containing all the elements that are common to both set A and set B called the _______________ of set A and set B. 4. The set of elements that belongs to set A but not in set B is called the _______________ of sets A and B. 5. A diagram that is used to represent sets is called _______________.

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is

What I Can Do

Activity 7: Create Me! Let’s Find Out: Venn Diagram Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. For sets U, A, and B, construct a Venn Diagram and place the

elements

in the proper regions b. Write the answer in your Mathematics notebook. Given: U = {iPhone, Blackberry, LG, Oppo, Vivo, Samsung, Nokia, Motorola, Sony} A = {iPhone, Blackberry, LG, Motorola, Oppo} B = {LG, Vivo, Nokia, Motorola}

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Assessment I. MULTIPLE CHOICE Directions: Read and answer the questions carefully. Write the answer in your Mathematics notebook. 1. What is the symbol used for intersection? A. ∪

C. ⊆

B. ∩

D. ∈

2. What is the complement of ∅? A. { } B. {1}

C. U D. {0}

3. What is the meaning of “the difference of P and Q”? A. The set of elements in P but not in Q. B. The set of elements is common to both P and Q. C. The set of elements in the universe that not in P and Q. D. The set of elements that is in P or Q or both P and Q. 4. What is the complement of {3, 4}, if U = {1, 2, 3, 4}? A. { }

C. { 1 }

B. { 0 }

D. { 1, 2 }

5. What is A ∪ B, if A= {0, 2, 3, 4, 9, 11} and B = {2, 3, 6, 8, 9, 10}? A. {0, 4 11} B. {2, 3, 6, 9}

C. {6, 8, 9, 10, 11} D. {0, 2, 3, 4, 6, 8, 9, 10, 11 }

6. If P = {a, b, c, d, e}, Q = {a, c, e, d, t } and R = { t, d, c, b, e }, then what is P∩ Q ∩ R? A. {a, c}

C. {c, d}

B. {a, c, e}

D. { c, d, e }

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7. Given U = {a, d, c, r, f, v, t, w}, H= { a, d, c, v } and J = { r, c, f }, what is H ∩ J’? A. {c }

C. { a, d, v, t }

B. {a, d, v}

D. {a, d, r, f, v, t }

8. What is (A ∪ B ) – C, if A = { 1, 2, 3, 4, 5, 6 }, B = { 2, 4, 5, 6 } and C = {1, 2, 4, 6 }? A. { }

C. { 3 }

B. { 2 }

D. { 3, 5 }

9. Which of the following represents the shaded area in the Venn Diagram below?

A. B’

C. B – A

B. A

D. A – B

10. What does the shaded region below represent?

A. (A ∪ B) ∩ C

C. A U (B – C)

B. A ∩ B ∩ C

D. A’ ∪ B ∩ C

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II. To solve numbers 11 – 15, use the given Venn Diagram below.

11. A ∩ B

13. A ∩ B’

12. A’

14. (A ∪ B)’

15. (A ∩ B )’

Additional Activities

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Activity 8: Find My Elements! Let’s Find Out: The Elements Let’s Use These Materials: Mathematics notebook and ballpen Let’s Do It This Way: a. Using the Venn diagram below, list the elements containing the set Write the answer in your Mathematics notebook. The first one is done for you!

1. U

Answer:

U = {red, black, blue, yellow, pink, maroon, orange, green, violet, white, indigo }

2. A

6. A’

10. A ‘∩ B

3. B

7. B’

4. A ∪ B

8. A – B

5. A ∩ B

9. B – A

Answer Key

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b.

What’s More

What’s New

(Activity 5)

(Activity 4)

2. B – D ‘ = { }

1. carrots, cabbage,

3. ( A ∪ B )’ – C

okra, squash,

= { 4, 6, 8 } 4. D – ( B ∩ A )

What’s In (Activity 3)

ampalaya, eggplant 2. 11

= { 2, 4, 6, 8 }

3. potato, raddish, string beans

What’s More

What’s In (Activity 1)

What I Know

(Activity 2)

1. SETS

1. B.

11. C.

2. ( B ∪ D ) ∩ C

2. UNIVERSAL SETS

2. C.

12. D.

3. NULL SET

3. C.

13. C.

4. EQUIVALENT SET

4. D.

14. D.

5. VENN DIAGRAM

5. D.

15. B.

6. FINITE

6. A.

7. ROSTER

7. D.

8. EQUAL SETS

8. C.

9. DISJOINT

9. B.

10. VERBAL

10. D.

= { 5, 7 } 3. A ∩ B ∩ C ={5} 4. C ∪ ( D ∩ A ) = { 1, 2, 3, 5, 7 } 5. ( A ∪ C ) ∩ B

= { 5, 7 }

Answer Key

Assessment

Additional Activity (Activity 8)

1. B.

11. { a, h }

2. C.

12. { f, g, r, d, p, m, z }

3. A.

13. { w, c, b, t }

4. D.

14. { p, m, z }

5. D.

15. { w, c, b, t, f, r, g, d, p, m, z }

2. A = { red, black, blue, yellow, pink, maroon } 3. B = { green, violet, orange, pink,

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maroon }

4. A ∪ B = { red, black, blue, yellow,

What I Can Do (Activity 7)

What I Have Learned (Activity 6) 1. complement 2. union 3. intersection 4. difference 5. Venn Diagram

References

BOOKS

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Von Anthony G. Toro, et.al. Smart in Math( Grade 7 ). ISA – JECHO PUBLISHING INC. 2017, pp. 10 – 13. Orlando A. Oronce and Marilyn O. Mendoza. e – math 7 ( K to 12 Worktext in Mathematics) Third Edition 2012. Rex Book Store, Inc. (RBSI) 2013. pp. 5 – 11. Gina Guerra and Catherine P. Vistro – Yu, Ed.D. Grade 7 Math Learning Guide. Department of Education (2013). pp. 7 – 18.

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