Unit 2. Lesson 2: The Language of Sets (Crobes, Maria Lorina B. and Escordial, Cristyflor M.

)
Learning Outcomes:
• Discuss basic properties of sets.
• Describe a set using roster method or rule method.
• Perform operations on sets.

One fundamental concept in mathematics is set theory. It is the branch of mathematics that studies sets or
the mathematical science of the infinite. George Cantor, a German mathematician introduced the study of sets in
the 19th century.

A set is a collection of well-defined distinct objects. Well-defined means that membership of a set is clear and
that there is a way to determine whether or not a given object belongs to a given set. A set has members or
elements. The objects that make up a set are called elements or members of the set.
Example. Identify the following as set or not a set.
a. Vowel letters in the English alphabet.
Solution. The set of vowels in the English alphabet are a,e,i,o,u
For you to tell if it is a set or not a set, let us go back to the definition of a set.
 Is it a collection of objects? Yes. A collection of vowels.
 Is the collection well-defined? Observe that for you to say that the collection is well-defined when you
can identify clearly the members and not members of the collection.
 Yes, it is well-defined because we can easily identify the members of this collection as letters a,e,i,o,u.
And we can clearly identify also those which are not members of this collection, i.e. consonant letters such
as c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z.
 Therefore, it is a set.
b. All beautiful people.
Solution. This is not a set. Why? For a collection of objects to be a set it should be that the collection is distinct
and well-defined. Observe that the word “beautiful” connotes different meanings dependent on who is the
beholder of the beauty. Beauty is relative. It can be beautiful to you but it may not hold true for others. Hence,
the members of this collection are not well-defined. You cannot precisely identify who will be part or members
of this collection. Therefore, it is not a set.
c. The set of counting numbers less than 10.
Solution. For this given collection to be a set, it should be that we can clearly identify distinct members of this
group of objects. Of course, this is a group of objects such that it is a counting number less than 10. Hence,
numbers like 1,2,3,4,5,6,7,8,9 are members of this group. While, numbers from 10,11,12,13,14,15,… are not
members of this collection. Thus, this is a set.
The table shows the symbol and its convention in set theory.
Symbol Conventions
∈ an object is an element of a set
∉ an object is not an element of a set
Uppercase letters to name sets
Lowercase letters to name elements of sets
{} a set or a list of elements of a set
𝜙 empty set
 U universal set
s.t. or / or : such that
 there exists, for some
 for all, for any
 set of Real Numbers
 set of Natural Numbers
ℤ set of Integers
ℤ+ set of Positive Integers
ℤ− set of Negative Integers
ℚ set of Rational Numbers

Example: For instance, let G be set of vowel letters. We write the set as G = {a, e, i, o, u}. The elements of
set G are a, e, i, o, and u. So, we write 𝑎 ∈ 𝐺, 𝑒 ∈ 𝐺, 𝑖 ∈ 𝐺, 𝑜 ∈ 𝐺, and 𝑢 ∈ G. However, b and c are not elements
of G. We write it as 𝑏 ∉ 𝐺 and 𝑐 ∉ 𝐺.
Methods of Describing/Naming a Set (Panopio et. al, 1999)
1. Roster Method or Tabulation Method
In this method, the elements of a set are listed and separated by commas and enclosed in braces.
Example 1: M is the set of whole numbers less than eight
Solution: M = {1, 2, 3, 4, 5, 6, 7}
Example 2: X is the set of all days in a week.
Solution: X = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
2. Rule Method or Set-Builder Notation
It indicates a set by enclosing in braces a descriptive phrase, and agreeing that those objects, and only those, which
have described property are the elements of the set.
Example 3: N = {2, 4, 6, 8}
Solution: N = {x | x is an even number less than 10}
This is read as, the set N contains the element x such that x is an even number less than 10.
Example 4: M = {January, February, March, April, May, June, July, August, September, October, November,
December}
Solution: M = {x | x is a month in a year}
This is read as the set of x’s such that x is a month in a year.

Kinds of Sets, Unit Set, Universal Set, Empty Set and Cardinal Number of a Set (Panopio et. al, 1999)
A set is said to be finite if it is possible to write down a complete list of all elements of the set. For instance, the
set of counting number less than or equal to 20 is a finite set. On the contrary, a set is said to be infinite if it is
impossible to write down all elements of the set. For example, the set of all positive integers is an infinite set.

A unit set is a set with only one element.

A unique set with no elements is called the empty set (or null set), and is denoted by 𝜙 or { }.
All sets under investigation are assumed to be contained in some large fixed set called the universal set, denoted
by the symbol U.
Example 5. Given sets J and K where:
J = {x| x is a positive integer between 11 and 13}
K = {5}.
Using the roster method, set J = {12}
The number of elements of J and K is one (1). Hence, J and K are unit sets.
Example 6: Let the set A = {x|x is a positive integer less than 1}
U= {x| x is a month with 33 days}
Since there is no positive integer that is less than 1 and also there is no month that has 33 days, therefore sets A
and U has no elements. It implies that A and U are empty sets.
The cardinal number of a set is the number of elements or members in the set. The cardinality of set A is denoted
by n(A).
Example 7. For instance, given the sets A = {a, b, c, d, e} and B = {x | x is an even number less than 20}
The cardinal number of A is 5 or n(A) = 5.
While set B has a cardinal number of 9 or n(B) = 9.
Subset, Proper Subset, Equal Set and Equivalent Set (Panopio et. al, 1999)
If A and B are sets, A is called a subset of B, written as 𝐴 ⊆ 𝐵, if and only if, every element of A is also
an element of B. Symbolically: 𝐴 ⊆ 𝐵 ↔ ∀𝑥, 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵. The number of subsets that can be formed from
any set is dependent on the number of elements in the set. Given a set A of n elements, we can form 2𝑛 subsets of
A. Any set is a subset of itself, as well as the empty set as a subset of any set.
A is a proper subset of B, written as 𝐴 ⊂ 𝐵, if and only if every element of A is in B but there is at least
one element of B that is not in A. In contrary, the symbol ⊄ denoted that it is not a proper subset.
Example 8. Suppose we have the following sets:
A = {2, 4, 6, 8, 10}
B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Is 𝐴 ⊆ 𝐵? Yes, since all elements of A is in B. That is 2 ∈ B, 4 ∈ B, 6 ∈ B, 8 ∈ B, 10 ∈ B.
Is 𝐴 ⊂ 𝐵? Yes, since all elements of A is in B but there are elements of B that are not in A. Like 1 ∈ B but
1  A, the same with 3, 4, 7 and 9.
Example 10. Let the following sets be:
M = {a, e, i, o, u}
N = {o, u, i, a, e}
Is 𝑀 ⊆ 𝑁? Yes, since all the elements in M are also found in N.
 Is 𝑀 ⊂ 𝑁? No, since all the elements in M are in N, However there is no element that is N which are not
in M.
A equals B, written as A = B, if and only if every element of A is in B and every element of B is in A.
Symbolically: 𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵) ∧ (𝐵 ⊆ 𝐴). On the other hand, A is equivalent to B, written as A ~ B, if they
have the same number of elements.

Example 11: We have the following sets: P = {1, 2, 3}; Q = {2, 3, 1}; R = {a, b, c}.
Is P = Q? Yes! They have the same elements.
Is Q = R? No. They don’t have the same elements.
Is P = R? No. They don’t have the same elements.
Is P ~ Q? Yes! They have the same number of elements
Is Q ~ R? Yes! They have the same number of elements
Is P ~ R? Yes! They have the same number of elements
Given a set S from universe U, the power set of S denoted by (𝑆), is the set of all subsets of S. For example, we
have A = {2, 4} so n(A) = 2. There are 2n = 2(2) = 4 subsets of A. The subsets of A are {2}, {4}, {2, 4}, and 𝜙.
The power set of A is (𝐴) ={{2},{4},{2,4},𝜙 }.
Operations on Sets (Panopio et. al, 1999)
The operations on sets are union of sets, intersection of sets, complement of a set, difference of two sets,
and symmetric difference of sets.
The union of sets A and B, denoted by 𝐴 ∪ 𝐵, is the set of all elements which belong either to A or to B, or to
both A and B. Symbolically, 𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵}.
The intersection of set A and B, denoted by 𝐴 ∩ 𝐵, is the set of all elements which belong to both A and B.
Symbolically, 𝐴 ∩ 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵}.
Example 7. We have the following sets: A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5}. What is 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵?
Solution: 𝐴 ∪ 𝐵 = {1, 2, 3, 4, 5, 6, 8, 10}
𝐴 ∩ 𝐵 = {2, 4}
The complement of A (or absolute complement of A), denoted by A’, is the set of all elements which belongs
to the universal set U but not in A. Symbolically, 𝐴′ ={𝑥|𝑥 ∈ 𝑼 ∧ 𝑥 ∉ 𝐴}. Suppose we have the universal set 𝑼 =
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and 𝐴 = {2, 4, 6, 8, 10}. The complement of A is denoted by 𝐴′ = {1, 3, 5, 7, 9}.
The difference of A and B (or relative complement of B with respect to A), denoted by A – B, is the set of all
elements that are in A but not in B. Symbolically, 𝐴 − 𝐵 = {𝑥|𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵}. Suppose we have the sets A = {a,
b, c, d, e} and B = {a, e, i, o, u}. The difference of A and B is given by 𝐴 − 𝐵 = {𝑏, 𝑐, 𝑑} while the difference
between B and A is given by 𝐵 − 𝐴 = {𝑖, 𝑜, 𝑢}. Note that A – B is not equal to B – A.
The symmetric difference of A and B, denoted by A ⊕ B, is the set consisting of all elements that belong to A
or to B, but not both A and B. Symbolically, 𝐴 ⊕ 𝐵 = {𝑥|𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝑥 ∉ (𝐴 ∩ 𝐵)}. Suppose we have the sets
A = {a, b, c, d, e} and B = {a, e, i, o, u}. The symmetric difference of A and B is given by 𝐴 ⊕ 𝐵 = {𝑏, 𝑐, 𝑑, 𝑖, 𝑜,
𝑢}.
Two sets are called disjoint (or non-intersecting) if and only if they have no elements in common. Symbolically,
𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑑𝑖𝑠𝑗𝑜𝑖𝑛𝑡 ↔ 𝐴 ∩ 𝐵 = 𝜙. Let A and B be subsets of the universal set U. For instance, A = {1, 2, 3}
and B = {4, 5, 6}. Since 𝐴 ∩ 𝐵 = 𝜙, then A and B are disjoint sets.
In the ordered pair (a, b), a is called the first component and b is called the second component. In general, (𝑎, 𝑏)
≠ (𝑏, 𝑎)
The Cartesian product of sets A and B, written as 𝐴 × 𝐵, is the set of all ordered pairs (a, b) for which the first
element is from set A and the second element from set B. Symbolically, 𝐴 × 𝐵 = {(𝑎, 𝑏)|𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵}. For
instance, we have sets A = {1, 2, 3} and B = (a, b}. The Cartesian product of A and B is 𝐴 × 𝐵 = {(1, 𝑎), (1, 𝑏),
(2, 𝑎), (2, 𝑏), (3, 𝑎), (3, 𝑏)}.

Let’s Try!
Let’s determine whether the statement is true or false.
1. 4  { 2, 3, 4}
2. 5  { 2, 3, 4}
3. {3,4}  { 1, 2, 3, 4, 5}
4. {2, 3, 4} = { 4, 3, 2, 4}
5. {2, 3, 4}  {4, 3, 2}
6. {3, 4}  {2, 3, 4}
7.   {2, 3, 4}

8.  = {0}

9. { }  {1}
10. { }  {1, 2, 3}