1. SETS

fasikülkümeler.indd 1 11.09.2015 08:57

Bu KİT, İSTEK Okulları Matematik Zümresi öğretmenleri tarafından hazırlanmıştır. 

Redaksiyonu, İSTEK Bilge Kağan Okulları Matematik Zümresince, 

uygulaması, İSTEK Bilge Kağan Okulları'nda Aslı Zeynep ALKAN tarafından yapılmıştır. 


SETS 

Georg Cantor was a German mathematician, best known as the inventor of set theory, which has become a 

fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between 

sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural 

numbers. Cantor's work between 1874 and 1884 is the origin of set theory. 

• Set is a well defined collection of objects or symbols which are called the elements of the set. 

A = {S, E, T} is given. S∈A but B∉A. 

• The Properties of a Set 

Basic Concepts Of Sets 

1. Each element is written only once. 

2. The order of the elements is not important. 

3. There must be a comma between any two elements in a set. 

4. The elements should be well defined. 

5. Each set is represented by a capital letter such as A, B, C, ... etc. 

Example 

A = {The vowels in our alphabet} = {a, e, ı, i, o, ö, u, ü} 

B = {Prime numbers between 6 and 30} = {7, 11, 13, 17, 19, 23, 29} 

C = {The letters in the word UNIVERSE} = {U, N, I, V, E, R, S} 

D = {The girls in 9 / A class} 

Representation of A Set 

We use three ways to represent sets: 

Listed Form 

This is the form in which we list all the members, separated by 

commas. 

A = {☎ , ✈ , ✉ , ✄ , ✏ , ❄ , ✒ , ✌ , ✍ , ✪} 

A 

.☎ .✈ .✉ .✄ .✏ 

.❄ .✒ .✌ .✍ .✪ 

Venn Diagram Form 

Sets can be shown in the regions enclosed in simple closed geometric figures such as; circles, squares, etc. The 

description of a set by a geometric region is called a Venn Diagram. Elements are marked as points. 

Set-builder Form (Described Form) 

We can establish a set builder form if there is a common property between elements. 

B = {months of the year beginning with the letter J} 

3 Mathematics KİT 9 

İSTEK YAYINLARI EPARGE 2015-2016 


Example: 

Let's write the set “the prime numbers which are greater than 3 and less than or equal to 23” in various set forms. 

P = {5, 7, 11, 13, 17, 19, 23} is in listed form. 

P 

. 5 . 7 . 17 

. 13 . 23 

. 11 . 19 

Set P in Venn 

diagram form. 

P = {x | 3 < x ≤ 23, x is a prime number} is in set-builder form. 

Exercises 

1. Determine whether the following are set or not 

A = {The smart boys of the class.} Set Not Set 

B = {The letters of MARMARA.} Set Not Set 

C = {The beautiful cities of TURKEY.} Set Not Set 

D = {The strongest men in the world.} Set Not Set 

E = { The odd integers.} Set Not Set 

2. Write the following sets in other forms if they can be written. 

a) A = {x | x = 2k, 0 ≤ k < 15 , k ∈ N} 

b) B = {Odd natural numbers less than 16} 

c) C = {1, 4, 9, 16, 25} 

d) 

D 

. 1 . 2 

. 3 . 4 

. 5 

e) E = {x : x.(x + 3).(x − 1) = 0 , x ∈ Z} 

f) F = {the letters of YEDİTEPE} 

3. Write the elements of the set B in a listed form if A = {x | x ≤ 8 , x ∈ N} and B = {x | x ∈ A , x + 1 < 8} 




The Number of Elements of A Set 

The notation n(A) is used to mean the number of elements in set A. 

A= {the planets in our solar system} n(A) = 8 

B = {Prime digits} n(B) = 4 

Universal Set and Empty Set 

Universal Set 

The set that contains all the elements being discussed is called the universal set shown by U or E. 

Example: 

If set J = {months of the year beginning with the letter J }, the universal set may be U = {months of the year} 

If set A = {positive even numbers less than 2011}, the universal set may be U = {2, 4, 6, 8, …} 

Empty Set 

A set which contains no elements is called an empty set or a null set denoted by { } or Ø 

A = {Integers divisible by 0} = Ø 

B= {Flying elephants} = Ø 

C= { } 

Example: 

Which one of the following sets is/are empty? 

a) {Ø} 

b) {0} 

c) {x : x2 = 9 and x = 5, x ∈ R} 

d) {n : n2 < 0 and n ∈ N} 

e) {x : x is even and a prime number} 


KİT 9 Mathematics 5 


ACTIVITY 

Mark the natural numbers from 1 to 9 on the number line. List the elements of this set and find the number of 

elements. 

Now mark the prime numbers 2, 3, 5, 7, 11 and 13 on the number line. How many more prime numbers can we mark? 

Finite and Infinite Set 

When we can write down, or count, all the members in a set, it is called a finite set. Otherwise it is an infinite set. 

A = {even natural numbers} is an infinite set. 

B = {months of the year beginning with the letter J} is a finite set. 

Z = {........., -3, -2, -1, 0, 1, 2, 3, ........} is an infinite set. 

Example: 

Find whether the following sets are finite or infinite. 

a) The set of integers divisible by 4 

b) Natural numbers between 5 and 120 

c) {k | k is a city in France} 

d) {n | n is a natural number between 2 and 3} 

e) {x|x∈Q and 

2 

≤ x ≤ 

5 

} 

3 7 

f) Natural numbers between any two different natural numbers a and b, which a < b. 

g) Rational numbers between any two different rational numbers x and y, which x < y. 

Subset and Equal Sets 

Subset 

If every element of a set A is an element of set B, then A is said to be a subset of B shown by A ⊂ B or B ⊃ A. 

Symbolically : A⊂ B = {x| x ∈ A ⇒ x ∈ B} 

B 

A 

Example: 

The set A = {a, b, c, d} is given. 

a) Write all the subsets of A with two elements. 

b) Write all the subsets of A. 




Example: 

The set A = { {3}, 4, {3,4}, 2, {1,2}, 1} is given. Write true or false for the following. 

a) {3} ⊂ A (…) b) {3} ∈ A (…) c) {{3}} ∈ A (…) d) {{3}} ⊂ A (…) 

e) {3,4} ⊂ A (…) f) {1,2} ⊂ A (…) g) {3, {3}, {3,4}} ⊂ A (…) 

h) {4, {3,4}, 2} ⊂ A (…) i) 2 ∈ A (…) j) {3,4} ∈ A (…) 

• Proper Subsets 

If a subset of a set is not equal to itself, it is called the proper subset. 

• Number of Subsets 

Let's write all the subsets of set A = {☺,☻,☹}: 

Ø , {☺} , {☻} , {☹} , {☺,☻} , {☺,☹} , {☹,☻} , {☺,☻,☹} 

There are 8 subsets of A. 

The number of proper subsets of A is 7. 

The number of improper subsets of A is 1 which is {☺,☻,☹} 

• If n(A) = n, the number of subsets of A is 2 n , 

the number of proper subsets of A is 2 n − 1 

Example: 

n(A) = 6 is given. Find the number of subsets and proper subsets of A. 

Example: 

n(A)= 3x + 1, n(B) = 2x − 3 are given. The number of subsets of A equals 64 times the number of subsets of B. 

a) Find n(A) + n(B). b) Find the number of proper subsets of A. 

Example: 

Find the number of proper subsets of the set M if it has 2 5n -8 

subsets for n(M) = n. 




Example: 

A= {a, b, c, d, e} is given. Find the number of 

a) subsets of A. 

b) subsets of A which don't contain c. 

c) subsets of A which don't contain a and b. 

d) subsets of A which contain e. 

e) subsets of A which contain d and e. 

f) subsets of A which contain c but not b. 

g) subsets of A which contain d and e but not a. 

h) subsets of A which contain c or d. 

Example: 

Let A = {a, b} and B = {a, b, c, d, e} are given. 

Write all the sets C satisfying the condition A ⊂ C ⊂ B. 

Example: 

Let set A = {x | x2 < 24, x ∈ N} and set B = {y | y < 10, y ∈ N} are given. 

Find the number of set C satisfying the condition A ⊂ C ⊂ B. 

• Properties of Subset 

1. Empty set is a subset of any set. 

2. Every set is a subset of itself. 

3. Every set is a subset of universal set. 

4. If A ⊂ B and B ⊂ C = A then A ⊂ C 

5. If A = B then A ⊂ B and B ⊂ A. 

6. If A ⊂ B then n(A) ≤ n(B) 

7. If B ⊂ A then A ∪ B = A and A ∩ B = B. 

Equal Sets 

M = {x | x is the natural number between 2 and 6} N = {x | 5 < x² ≤ 28 and x ∈ N} 

As you can see the elements of N and M are same. These sets are called equal sets and written as M = N. 

If M=N then M ⊂ N and N ⊂ M 

If M ⊂ N and N ⊂ M then M=N 




Exercises 

1. Determine whether the groups below are sets or not. 

a) Easy lessons. 

b) The cities neighbour to Ankara. 

c) The lakes in Turkey. 

d) The beautiful girls of İzmir. 

2. Fill in the blanks with ∈ or ∉. 

A= {T, U, R, K, E, Y} 

a) T.............A b) Y.............A 

c) F.............A d) B.............A 

3. 

. İ 

B 

The set B is given. 

Which of the followings are true? 

. S . T 

I. İ ∈ B 

II. 

K ∈ B 

. E 

. K 

III. B = {The letters of İSTEK.} 

IV. B = {İ, S, T, E, K} 

V. n(B) = 4 

4. C = {k | -2 ≤ k

7. Let F = {1, 2, 3, 4, 5, 6, 7} write all the subsets with two elements which contain 5. 

8. Let G = {a, b, c, d, e} write all the subsets with two elements. 

9. Let H = {Prime digits} find the number of subsets of H with two elements. 

10. Let I = {a, b, c, d, e, f} find the number of all subsets of I with three elements having the element a but not having b. 

11. Let the number of all subsets of the set J be x and the number of proper subsets of J be y. If x + y = 63 then find the 

number of the elements of set J. 

12. K = {1, 2, 3, 4, 5, 6, 7} find the number of 

a) subsets of K. 

b) subsets of K which don't contain 1. 

c) subsets of K which don't contain 2 and 3. 

d) subsets of K which contain 4. 

e) subsets of K which contain 5 and 6. 

f) subsets of K which contain 7 but not 1. 

g) subsets of K which contain 2 and 3 but not 3. 

h) subsets of K which contain 4 or 5. 




Example: 

Show that A = {x | 4 < x < 12, x = 2k + 1, k ∈ N} and B = {5, 7, 9, 11} are equal sets. 

Example: 

Divide the given figure by drawing 3 straight lines, so that you get 

4 different trees in each region. 

Disjoint Sets 

If two sets have no common elements then they are called disjoint sets. 

Example: 

Show that A = {1, 2, 3, 4} and B = {a, b, c, d} are disjoint sets. 




SELF TEST 

1. A = {x : 1 ≤ x ≤ 10 , x ∈ Z} 

B = {(x,y) : x + y < 15, x, y ∈ A} 

For the given sets above, find the number of elements of set B. 

A) 10 B) 14 C) 65 D) 79 E) 85 

2. A set has three conditions: 

I. The elements of the set are consecutive positive integers. 

II. The set has at least two elements. 

III. The sum of the elements of the set is 45. 

Find the number of sets satisfying these conditions. 

A) 1 B) 2 C) 3 D) 4 E) 5 

3. Which one of the following is true for the set A = {1, {1}, {2}, {2,3}} ? 

A) 2 ∈ A B) {2} ⊂ A C) {3}∈ A D) {1, {1}} ⊂ A E) {2, 3} ⊂ A 

4. Which one of the following is true for the sets A = {ATATÜRK} and B = {TÜRK} ? 

A) B ⊂ A B) n(A) = n(B) C) A = B D) A ⊂ B E) n(A) > n(B) 

5. Which one of the following is not an empty set? 

A) {x : x2 < 0, x ∈ R} 

B) {x : 2x + 8 = 0, x ∈ Z } 

C) The deadless animals. 

D) {x : x2 < x, x ∈ Z} 

E) {Ø} 




6. If the number of subsets of set A including the elements 1 and 2 is 32, then find the number of subsets of A. 

A) 32 B) 64 C) 128 D) 256 E) 512 

SELF TEST 

7. How many subsets of set A = { 0, 1, 2, 3, 4, 5, 6 } don’t have any even numbers? 

A) 8 B) 16 C) 24 D) 32 E) 64 

8. When we add two new elements to a set the number of all subsets increases by 96. Find the number of elements 

in this set. 

A) 4 B) 5 C) 6 D) 7 E) 8 

9. How many subsets of set A = {a, b, c, d, e} have at most one of the elements “d” or “e” ? 

A) 32 B) 24 C) 18 D) 16 E) 8 

10. The set A has 3 elements and B has b elements. The sum of the number of all subsets of a set A and the number 

of all proper subsets of the set B is 39. Find the value of b. 

A) 3 B) 4 C) 5 D) 6 E) 7 




11. Let A and B be two sets. If A ⊂ B, n(A) = 4 and n(B) = 6, then how many subsets of set B have 2 elements of set 

A? 

A) 24 B) 40 C) 44 D) 48 E) 64 

SELF TEST 

12. How many subsets of set A = { 1, 2, 3, 4, 5 } have at most one odd number? 

A) 4 B) 8 C) 16 D) 24 E) 32 

13. Let B = {1, 2, 3, x, y}, find the number of subsets with 3 elements which contain 1. 

A) 9 B) 8 C) 7 D) 6 E) 5 

14. Let C = {a, b, c, d, e, f}, find the number of subsets of C which contain f but not b. 

A) 2 B) 4 C) 8 D) 16 E) 32 

15. Find the number of proper subsets of the set A if it has 2⁴ⁿ - ¹² subsets for n(A) = n. 

A) 63 B) 32 C) 31 D) 16 E) 15 

Answer key: 1 D - 2 D - 3 D - 4 B - 5 E - 6 C - 7 A - 8 B - 9 B - 10 C - 11 A - 12 C - 13 D - 14 D - 15 E 




SET OPERATIONS 

ACTIVITY 

A firm is looking for people to work for the sales and public 

relations departments. In terms of the applications 

Ahmet, Mustafa and Erol are assigned to work in the 

sales department, Funda and Gamze are assigned to 

work in the public relations department, Kübra, Mert and 

Esin are assigned to work in both the sales and public 

relations departments. 

✓ Draw a Venn diagram to represent this situation. 

✓ Just write down the sales department staff. 

✓ Just write down the public relations department staff. 

✓ Write down either sales or public relations departments 

staff. 

✓ Write down both sales and public relations departments 

staff. 

Intersection and Union of Sets 

Intersection of Sets 

The intersection of two sets A and B is the set of all elements which belong to both A and B. It is shown by A∩ B. 

Symbolically : A ∩ B A 

B 

A ∩ B 

A ∩ B= {x| x ∈A and x ∈ B} 

If A and B are disjoint then A ∩ B = Ø 

A 

. 3 

. 1 

B 

. a 

. 2 

Example: 

A = {1, 2, 3, a}, B = {a, b, c, d} and C = {d, e, f} are given 

a) A ∩ B = 

b) A ∩ C = 

c) B ∩ C = 




Union of Sets 

The union of two sets A and B is the set of all elements which belong to A or B. It is shown by A ∪ B. 

Symbolically: A∪ B= {x| x∈A or x∈B} 

A 

AUB 

B 

• The Number of Elements of A∪B 

a. If A and B are disjoint 

A 

B 

. 1 

. 3 

. a 

. 2 

A = {1, 2, 3} ⇒ n(A) = 3 

B = {a} ⇒ n(B) = 1 

A ∪ B = {1, 2, 3, a} ⇒ n(A ∪ B) = 4 

n(A ∪ B) = n(A) + n(B) 

b. If A and B have common elements. 

A 

. a 

. 3 

. 1 

. . 

2 

b 

B 

A = {1, 2, 3} ⇒ n(A) = 3 

B = {a, b, 1} ⇒ n(B) = 3 

A ∪ B = {1, 2, 3, a, b} ⇒ n(A ∪ B) = 5 

n(A ∪ B) = n(A) + n(B) − n(A ∩ B) 

• The properties of union and intersection of sets 

1. Idempotent property 

A ∪ A = A 

A ∩ A = A 

2. Commutative Property 

A ∪ B = B ∪ A 

A ∩ B = B ∩ A 

3. Associative Property 

(A ∪ B) ∪ C = A ∪ (B ∪ C) 

(A ∩ B) ∩ C = A ∩ (B ∩ C) 

4. Distributive Property: 

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) 

5. a) A ∪ Ø = A b) A ∪ U = U 

A ∩ Ø = Ø 

A ∩ U = A 

6. a) If A ∩ B = Ø then n(A ∪ B) = n(A) + n(B) 

b) If A ∩ B ≠ Ø then 

n (A ∪ B) = n(A) + n(B) − n(A ∩ B) 

c) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C) 




Example: 

A = {1, 2, 3}, B = {2, 3, 4} and C = {1, 2, 3, 4, 5} are given 

Find the following and show them in Venn diagram. 

A⋃B = 

A⋃C = 

B⋃C = 

Example: 

U 

According to the Venn diagram find the fallowing. 

A 

. 1 

. + 

. 2 

B 

a) U = 

. a 

. ∞ 

. b 

b) A = 

. x . y 

c) B = 

d) A∩B = 

e) A⋃B = 

Example: 

A = {a, b, c, d}, B = {c, e, f} and C = {b, d, g, h} 

a) Write the following by listed form. 

A⋃B = 

A∩B = 

B∩C = 

B⋃C = 

A∩C= 

A⋃C = 

b) Show the distributive property by using given sets 

A⋃(B∩C) = (A⋃B)∩(A⋃C) 

A∩(B⋃C) = (A∩B) ⋃ (A∩C) 




Example: 

n(A) = 4, n(B) =6 and n(A∩B) =3 is given. 

Find n(A∪B). 

Example: 

n(A) = 6, n(B) =8 then find all possible values of n(A∩B). 

Example: 

n(A) = 4. n(B) and n(A∩B) + n(A∪B) =20. 

Find n(A). 

Example: 

Let n(A∪B) =13, n(A∪C) =19 and n(B∪C) =10 

n(A) + n(B) + n(C). 

if n(A∪B∪C) =17 + n(A∩B∩C), then find the value of 




Exercises 

1. Find the following if A= {x| −4 < x < 4, x∈Z} and B= {x| x2 ≤ 16, x∈Z} 

A = 

B = 

A ∩ B = 

A ∪ B = 

2. Find A ∩ B and A ∪ B if 

A = [−4, 4] and B= [−2, 6) 

3. Find A ∩ B and A ∪ B if 

A= (−4, 2] and B= [−2, 4) 

4. A= {1, 2, 3, 4,...., 100}, how many elements of the set A are divisible by 

a) 3 

b) 4 

c) both 3 and 4 

d) 3 or 4 

e) 3 but not 4 

f) 4 but not 3 

5. A = {x ; 24 ≤ x ≤ 85, x∈N}. 

How many elements of the set A are divisible by 

a) 2 

b) 3 

c) 2 and 3 

d) 2 or 3 

e) 2 but not 3 

f) 3 but not 2 

6. If A= {x| x < 100 , x = 7k , k∈Z } and B= {x| x < 221, x = 3k, k∈Z } 

+ + 

Find n(A ∪ B). 

7. If A = {x| 50 < x < 200 , x = 3k , k∈Z} and B = {x| 30 < x ≤ 135 , x = 5k , k∈Z} 

Find n(A ∩ B). 

8. A= {x ; 40 < x ≤ 240 , x = 3n , n∈N} and B = {y ; 50 < y < 300 , y = 4k , k∈N} are given. 

Find n (A ∪ B). 

9. If A ∪ B = {a, b, c, d, e, f} and A ∪ C = {a, b, d, 1, 2, 3} 

Find A ∪ (B ∩ C). 

10. If A ∩ B = {1, 2, 3, 4} and A ∩ C = {1, 3, 4, 5, 6} 

Find A ∩ (B ∪ C). 




ACTIVITY 

During a blood transfer the receiver needs all the antigens’ of the donor. A person may have either one, two or all of the 

A, B, Rh antigens or none of them. In terms of this situation, 8 possible different blood groups are shown in the Venn 

diagram below. Here set E includes all of these possible blood groups. 

A 

B 

Blood Groups 

ARh- 

ARh+ 

ARh- ABRh- BRh- 

BRh- 

BRh+ 

ABRh+ 

ABRh- 

ARh+ BRh+ 

ABRh+ 

0Rh- 

0Rh- 

0Rh+ 

0Rh+ 

Rh 

For example; a person who belongs to ARh− group has got A antigen and he hasn’t got B and Rh antigens, a person 

who belongs to 0Rh+ group has got Rh antigen and he hasn’t got A and B antigens, a person who belongs to ABRh− 

group has got A and B antigens and he hasn’t got Rh antigen. 

In each set below find out which of the 8 blood groups are included. 

A ∩ Rh = {ARh+, ABRh+} 

a) A ∩ B b) A ∪ Rh c) A ∪ B d) A ∩ Rh e) B ∪ Rh f) (A ∪ B) ∩ Rh g) A ∪ B ∪ Rh 

Complement of A Set 

The set of all elements of the universal set 

that are not in A is called the complement 

of A denoted by A' or A. 

Symbolically : A' = {x| x ∈ U and x ∉ A} 

U 

A΄ 

A 

Example: 

If A = {1, 3, 5, 7} and universal set U= {x| x is numeral} then find A' ? 




̸ 

̸ 

• The Properties of Complement of A Set 

1. A ∪ A΄ = U 

A ∩ A΄ = Ø 

2. (A΄)΄ = A 

3. U ΄ = Ø , Ø ΄ = U 

4. (A ∪ B)΄ = A΄ ∩ B΄ 

(A ∩ B)΄ = A΄ ∪ B΄ 

5. n(A) + n(A΄) = n(U) 

Example: 

1. U = {x | x is digit} is universal set A = {x | x is an even numeral} and B = {x | x is a prime numeral} 

a) Show the sets in Venn diagram 

b) n(U)= n(A)= n(B)= n(A')= and n(B')= 

2. U = {1, 2, 3, 4, 5, 6} , A = {1, 2, 3} and B = {3, 4, 5}. Find, 

B ∩ B΄ 

A΄ 

(A ∩ B)΄ 

(A ∪ B)΄ 

A΄ ∩ B 

A ∪ A΄ 

B΄ 

A ∪ B΄ 

3. Find A' if A = {x | x ∈ N and it is divisible by 2} and U = {natural numbers less than 15} 

4. If U = [1, 8) and A = [1, 4], find A' 

5. If U = [−1, 8) , A = [3, 6] and B = (1, 4), find B ∩ A' 

6. Simplify (B' ∪ A') ∪ [A ∩ (B ∪ A')] 

Difference of Two Sets 

The difference of the sets A from the set B is the set of elements of the set A which do not belong to the set B. It is shown 

by A \ B or A − B. Symbolically : A \ B = {x : x ∈ A and x ∉ B} 

The difference of the sets B from the set A is the set of elements of the set B which do not belong to the set A. It is shown 

by B \ A or B − A. Symbolically : B \ A = {x : x ∈ B and x ∉ A} 

A 

B 

A B 

A∩B 

B A 




• The properties of A \ B 

1. A \ A = ∅ and A \ A΄ = A 

2. A \ ∅ = A and A \ U = ∅ 

3. U \ A = A΄ and U \ A΄ = A 

4. If A ≠ B then, A \ B ≠ B \ A 

5. A \ B = A ∩ B΄ 

6. n(A ∪ B) = n(A \ B) + n(B \ A) + n(A ∩ B) 

7. A∆B= (A \ B) ∪ (B \ A) 

Example: 

1. Find A \ B and B \ A if A = {a, b, c, d, e} and B = {b, d, e, f, g} 

A \ B = 

B \ A = 

2. If A=[−2, 5) and B= [1, 9], find A \ B. 

3. If A= (−2, 5] , B= [−1, 6] and C= [−3, 4], find C \ (A ∩ B). 

4. Simplify the following by using properties 

a) (A ∪ ∅) ∪ (A ∩ ∅)΄ 

b) (A − B) ∪ (A − B΄) 

c) (B − A) ∪ (B΄ − A) 

5. If n(A \ B) = x2 , n(A ∩ B) = 4, n(A) = 4x and n(A΄ ∩ B) = 4 , find n(A ∪ B). 

6. A = {Doctors in the group} 

B = {Men in the group} 

C = {Women in the group} 

D = {People with glasses in the group} 

Find (B ∩ D) − (A ∪ C). 

7. Define the shaded regions below. 

A 

B A 

B A 

B A 

a) b) c) d) 

B 

C 

C 

C 

C 




ACTIVITY 

Let A = {Türkiye, Bulgaria, Greece, Macedonia} and B = {Ankara, Athens, Sofia, Uskup} 

Write the elements of the relation between the states and their capitals as (Bulgaria, Sofia). 

Ordered pair is the pairing of elements in a stated order written as 

(a,b). Ordering is important. (a,b) ≠ (b,a) 

(2,3) ; (−5,0) ; ... are ordered pairs. 

The Equality of Ordered Pairs 

(a, b) = (c, d) if and only if a = c and b = d 

Ordered Pair 

first 

component 

(a, b) 

second 

component 

Example: 

If (a + b , 3) = (17, a – 2) then find b. 

Example: 

If (x + y, 5x ) = (3x, 25) then x + y = ? 

Example: 

If (x3, x + 2) = (y, −1) then y = ? 

Cartesian Product 

The cartesian product of the sets A and B shown as A × B is the set of all ordered pairs (x, y) such that 

x ∈ A and y ∈ B. A × B = {(x, y) : x ∈ A and y ∈ B} 

Example: 

If A = {1, 2} and B = {a, b}. Write, 

a) A × B = 

b) B × A = 




Example: 

A = {a, b} and B = {1, 2, 3}. 

a) Write A × B in listed form. 

b) Show B × A in Venn diagram. 

c) Draw the graph of B × B 

Properties of Cartesian Product : 

✓ A ≠ B ⇒ A × B ≠ B × A 

✓ A × ∅ = ∅ × A = ∅ 

✓ A × B × C = A × (B × C) = (A × B) × C 

✓ A × A = A2 , A × A × … × A = Aⁿ 

✓ R × R = R2 

✓ A × (B ∩ C) = (A × B) ∩ (A × C) 

✓ A × (B ∪ C) = (A × B) ∪ (A × C) 

✓ A × (B − C) = (A × B) − (A × C) 

Example: 

A = {1, 2, 3} and B = {4, 5}. 

Show A × B in listed form, Venn diagram and graph. 

Example: 

Let A = {1, 2, 3}, B = {2, 4} and C = {4, 5}. 

a) A × (B ∪ C) 

b) (A × B) ∪ (A × C) 

c) A × (B ∩ C) 

d) (A × B) ∩ (A × C) 

Write in listed form, then compare these sets. 




Example: 

If A = {1, 2, 3}, A ⊂ B, which one of the following is not an element of A × B. 

A) (1,4) B) (1,5) C) (5,2) D) (2,1) E) (1,2) 

Example: 

If A × B = {(a,1),(a,2),(b,1),(b,2)} , B × C = {(1,1),(1,3),(1,5),(2,1),(2,3),(2,5)} then A ∪ C = ? 

Number of Elements of Cartesian Product: 

If n(A) = m, n(B) = n, then n(A × B) = n(A).n(B) = m.n 

Example: 

A = {1, 2, 3} and B = {a, b} 

A × B = 

B × A = 

A × A = 

B × B = 

n(A × B) = 

n(B × A) = 

n(A × A) = 

n(B × B) = 

Example: 

If n[ (A × B) ∪ (A × C) ] = 52 , n(B ∪ C) = 13 then n(A) = ? 

Example: 

If A ∩ B = {a,b}, C = {0,1,2,3} then n( (A × C) ∩ (B × C) ) = ? 

Example: 

If A = {a,b,c,d}, B = {b,c,d,e,f,g,k,l}, C = {c,d,e,r} , then n[(A × B) ∩ (A × C)] = ? 




• REMARK 

If in a classroom there are students who speak only English, only 

German, both of these languages or none of these languages, then 

b: English and German 

a+b+c: English or German 

a: only English 

c: only German 

a+c: only English or only German 

a+b+c: at least one of these languages 

a+c+d: at most one of these languages 

E 

a b c 

G 

d 

SET PROBLEMS 

1. In a class of 24 students, 16 students study mathematics, 12 study science and 5 study neither. How many of them 

study both? 

2. In a group of 500 people 32% of them know language A, 49% of them know language B. At most how many of them 

may know neither of them? 

3. Out of 45 students in a class 15 select English, 13 don't select Math, 6 select none of them. 

How many of them select one subject only? 

4. n(A) =10, n(B)=9 and n(A⋃B) =15, find the number of elements of A−B. 

5.The number of all the subsets of A−B is 16 and the proper subsets of B−A is 7. If the number of elements of A⋃B is 20 

then find n(A∩B). 

6. In a class of 200 students of a certain high school, the official records indicate that 80 had taken Physics, 90 had taken 

Biology, 55 had taken Chemistry, 32 had taken both Biology and Physics, 23 had taken both Chemistry and Physics, 16 

had taken both Biology and Chemistry and 8 had taken all three subjects. 

a) How many students had not taken any of these three courses? 

b) How many students had taken Physics and Biology but not taken Chemistry? 




SELF TEST 

1. A = {x : −2 ≤ x ≤ 6, x ∈ R} 

B = {x : −6 < x < 2, x ∈ R} 

Find A ∩ B for the given sets above. 

A) [−4,4] B) [−2,6) C) [−2,2) D) [2,4) E) [−6,6) 

2. Let A and B be the subsets of the universal set Z (integers). If A' = {1, 2, 3} and B' = {2, 3, 4}, 

then find the number of elements of (A ∩ B)' . 

A) 2 B) 3 C) 4 D) 5 E) 6 

3. Let m be a positive integer. If n(A) = m + 5, n(B) = 2m – 5, n(A ∩ B) = m + 1 and 

n(A \ B) = 2.n(B \ A), find the number of proper subsets of B. 

16 11 8 3 2 

A) 2 − 1 B) 2 − 1 C) 2 − 1 D) 2 − 1 E) 2 − 1 

4. A = {a, b, c, d, e, f} 

B = {b, d, e, f, m, r, k} 

C = {d, m, f, b} 

Find the number of elements in the shaded region. 

A) 1 B) 2 C) 3 D) 4 E) 5 

A 

C 

B 

5. Which one of the following does not define the shaded region in the given Venn diagram? 

A) (B \ C) ∩ A B) (A ∩ B) \ C C) (A \ C) ∩ B D) (A ∩ B) \ (C ∩ B) E) (A ∩ B) ∩ C 

A 

C 

B 




SELF TEST 

6. Which one of the following shaded regions shows the people who play one of the games football, volleyball or 

basketball ? 

A) 

F 

B F B F B F B F 

B 

B) 

C) 

D) 

E) 

V V V V V 

7. The color of some of Gaye’s pencils and all of Merve’s pencils is the same. The color of Özge’s pencils is 

completely different from Merve’s ones. The color of some of Gaye’s pencils and some of Özge’s pencils is the 

same. If G is the set of Gaye’s pencils, Ö is the set of Özge’s pencils and M is the set of Merve’s pencils, then 

which one of the following is true? 

A) 

M 

Ö 

G 

B) 

G 

Ö 

C) 

G 

M 

Ö 

D) 

M 

G 

Ö 

E) 

G 

Ö 

M 

M 

8. All people of a group eat 1, 2, 3 or 4 apples in a day. If the number of people who eats 2 apples is equal 

to the number of people who eats 3 apples, and the total number of eaten apples is 36 more than the total 

number of people in the class, find the number of people who eats at least 3 apples? 

A) 12 B) 18 C) 24 D) 30 E) 36 

9. A group of people can speak one language of either French, English or German at least. The people 

speaking French can speak English too. 3 of them can speak German only, 5 of them can speak English only, 

6 of them can speak German and 7 of them can speak two languages at least. How many people don't speak 

German? 

A) 7 B) 8 C) 9 D) 10 E) 11 

10. In a group of 63 people who play either the guitar, piano or drum which is 6 times the number of people 

who play none of them. If 40 can play two instruments at least, how many people can play one instrument 

only? 

A) 9 B) 14 C) 20 D) 30 E) 36 




11. In a class of students everyone is successful in at least one of courses. 45% of them are successful in Mathematics, 

85% of them in Geometry. If 6 students are successful in both of them, find the number of students which are 

successful in Mathematics? 

A) 6 B) 9 C) 12 D) 18 E) 24 

SELF TEST 

12. In a class, everyone is successful in at least one of courses. 20% of the students are successful in both History and 

Math. The number of students who are successful only in History is one third of the number of students who are 

successful only in Math. If the number of students who are successful in both is 10, find the number of the students 

who are successful only in Math. 

A) 12 B) 18 C) 24 D) 30 E) 36 

13. The students who passed Physics also failed Chemistry. 10% of the class passed Chemistry only, 5% of the class 

passed Physics only and 20% of the class passed Maths only. 55% of the class passed Physics or Chemistry, what percent 

of students passed Math? 

A) 20 B) 30 C) 40 D) 50 E) 60 

14. 60% of students in a class passed both Mathematics and Physics, 80% of them passed Mathematics and 70% of 

them passed Physics. 4 students failed Mathematics and Physics. How many students passed only Mathematics? 

A) 4 B) 6 C) 8 D) 10 E) 12 

15. In a group of people, 18 have glasses, 30 don't have glasses. If 36 are old or don't have glasses, how many of the 

young people have glasses? 

A) 12 B) 18 C) 20 D) 30 E) 36 

Answer key: 1 C - 2 C - 3 B - 4 B - 5 E - 6 A - 7 C - 8 A - 9 C - 10 B - 11 B - 12 D - 13 E - 14 C - 15 A 




ACTIVITY 

Find the related words about SETS in the puzzle below. The words can be forwards or backwards, vertical, horizontal or 

diagonal. Circle each word separately but keep in mind that letters may be used in more than one word. 




NOTE 




İSTEK ÖZEL ACIBADEM OKULLARI 

Acıbadem Mah. Bağ Sok. No: 6 34718 Acıbadem-Kadıköy/İSTANBUL Tel: 0216 325 30 75 

İSTEK ÖZEL ANTALYA YEDİTEPE KOLEJİ 

Tarım Mah. 1613 Sok. Koçtaş Arkası No: 21 ANTALYA Tel: 0242 312 43 43 

İSTEK ÖZEL ATANUR OĞUZ OKULLARI 

Balmumcu Mah. Gazi Umur Paşa Sok. No: 26 34349 Beşiktaş/İSTANBUL Tel: 0212 211 34 60 - 61 

İSTEK ÖZEL BELDE OKULLARI 

Kuzguncuk Mah. Rasimağa Sok. No: 7/4 34664 Üsküdar/İSTANBUL Tel: 0216 495 96 23 

İSTEK ÖZEL BİLGE KAĞAN OKULLARI 

Şenlikköy Mah. Florya Cad. No: 2 34153 Florya-Bakırköy/İSTANBUL Tel: 0212 663 29 71-73 

İSTEK ÖZEL KAŞGARLI MAHMUT OKULLARI 

50. Yıl Mah. Eski Edirne Asfaltı No: 512 34110 Sultangazi/İSTANBUL Tel: 0212 594 22 69 

İSTEK ÖZEL KEMAL ATATÜRK OKULLARI 

Tarabya Mah. Tarabya Bayırı Cad. No: 60 34457 Sarıyer/İSTANBUL Tel: 0212 262 75 75 / 262 42 57 

İSTEK ÖZEL SEMİHA ŞAKİR OKULLARI 

Caddebostan Mah. Bağdat Cad. No: 238/1 34730 Kadıköy/İSTANBUL Tel: 0216 360 12 18 - 356 90 92 

İSTEK ÖZEL ULUĞBEY OKULLARI 

Atalar Mah. Akgün Sok. No: 23 34862 Kartal/İSTANBUL Tel: 0216 488 13 08 

İSTEK OKULLARI GENEL MÜDÜRLÜĞÜ 

Acıbadem Mah. Bağ Sok. No: 8 34718 Acıbadem - İstanbul 0216 326 34 15 www.istek.k12.tr

1. SETS

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