# Sets and their subsets Contributed by: This pdf includes the following topics:-
Set
Set Notation
Ellipsis
The cardinality of a Set
Symbols commonly used with Sets
Number of Proper Subsets
Complement of a Set
1. Sets and Subsets
Set - A collection of objects. The specific objects within the set are called the
elements or members of the set. Capital letters are commonly used to
name sets.
Examples: 𝑆𝑒𝑡 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} 𝑜𝑟 𝑆𝑒𝑡 𝐵 = {1, 2, 3, 4}
Set Notation - Braces { } can be used to list the members of a set, with each
member separated by a comma. This is called the “Roster Method.” A
description can also be used in the braces. This is called “Set-builder”
notation.
Example: Set A: The natural numbers from 1 to 10. Roster Method
Members of A: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Set Notation: 𝐴 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Set Builder Not.: {𝑥 𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑓𝑟𝑜𝑚 1 𝑡𝑜 10}
Ellipsis - Three dots (…) used within the braces to indicate that the list continues
in the established pattern. This is helpful notation to use for long lists or
infinite lists. If the dots come at the end of the list, they indicate that the
list goes on indefinitely (i.e. an infinite set).
Examples: Set A: Lowercase letters of the English alphabet
Set Notation: {𝑎, 𝑏, 𝑐, … , 𝑧}
Cardinality of a Set – The number of distinct elements in a set.
Example: Set 𝐴: The days of the week
Members of Set A: Monday, Tuesday, Wednesday,
Thursday, Friday, Saturday, Sunday
Cardinality of Set 𝐴 = 𝒏(𝑨) = 7
Equal Sets – Two sets that contain exactly the same elements, regardless of the
order listed or possible repetition of elements.
Example: 𝐴 = {1, 1, 2, 3, 4} , 𝐵 = {4, 3, 2, 1, 2, 3, 4, } .
Sets 𝐴 𝑎𝑛𝑑 𝐵 are equal because they contain exactly the same
elements (i.e. 1, 2, 3, & 4). This can be written as 𝑨 = 𝑩.
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2. Equivalent Sets – Two sets that contain the same number of distinct elements.
Example: 𝐴 = {𝐹𝑜𝑜𝑡𝑏𝑎𝑙𝑙, 𝐵𝑎𝑠𝑘𝑒𝑡𝑏𝑎𝑙𝑙, 𝐵𝑎𝑠𝑒𝑏𝑎𝑙𝑙, 𝑆𝑜𝑐𝑐𝑒𝑟} Both Sets have 4
𝐵 = {𝑝𝑒𝑛𝑛𝑦, 𝑛𝑖𝑐𝑘𝑒𝑙, 𝑑𝑖𝑚𝑒, 𝑞𝑢𝑎𝑟𝑡𝑒𝑟} elements
𝑛(𝐴) = 4 𝑎𝑛𝑑 𝑛(𝐵) = 4
𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑆𝑒𝑡𝑠, 𝑚𝑒𝑎𝑛𝑖𝑛𝑔 𝑛(𝐴) = 𝑛(𝐵).
Note: If two sets are Equal, they are also Equivalent!
Example: 𝑆𝑒𝑡 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} 𝑆𝑒𝑡 𝐵 = {𝑑, 𝑑, 𝑐, 𝑐, 𝑏, 𝑏, 𝑎, 𝑎}
Are Sets A and B Equal? Sets A and B have the  Yes!
exact same elements!
{𝑎, 𝑏, 𝑐, 𝑑}
Are Sets A and B Sets A and B have the
Equivalent? exact same number of  Yes!
distinct elements!
𝑛(𝐴) = 𝑛(𝐵) = 4
The Empty Set (or Null Set) – The set that contains no elements.
It can be represented by either { } 𝑜𝑟 ∅.
Note: Writing the empty set as {∅} is not correct!
Symbols commonly used with Sets –
∈ → 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑎𝑛 𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑠 𝑎𝑛 𝒆𝒍𝒆𝒎𝒆𝒏𝑡 𝑜𝑓 𝑎 𝑠𝑒𝑡.
∈ → 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑎𝑛 𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑠 𝒏𝒐𝒕 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑠𝑒𝑡.
 → 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑎 𝑠𝑒𝑡 𝑖𝑠 𝑎 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑠𝑒𝑡.
 → 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑎 𝑠𝑒𝑡 𝑖𝑠 𝑎 𝒑𝒓𝒐𝒑𝒆𝒓 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑠𝑒𝑡.
∩ → 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑡ℎ𝑒 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝑜𝑓 𝑠𝑒𝑡𝑠.
∪ → 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑠 𝑡ℎ𝑒 𝒖𝒏𝒊𝒐𝒏 𝑜𝑓 𝑠𝑒𝑡𝑠.
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3. Subsets - For Sets A and B, Set A is a Subset of Set B if every element in Set A is
also in Set B. It is written as 𝑨  𝑩.
Proper Subsets - For Sets A and B, Set A is a Proper Subset of Set B if every
element in Set A is also in Set B, but Set A does not equal Set B. (𝑨 ≠ 𝑩)
It is written as 𝑨  𝑩.
Example: 𝑆𝑒𝑡 𝐴 = {2, 4, 6} 𝑆𝑒𝑡 𝐵 = {0, 2, 4, 6, 8}
{2, 4, 6}  {0, 2, 4, 6, 8} and {2, 4, 6}  {0, 2, 4, 6, 8}
Set A is a Subset of Set B Set A is a Proper Subset of Set B
because every element in A is because every element in A is also
also in B. 𝑨  𝑩 in B, but A ≠ 𝐵. 𝑨  𝑩
Note: The Empty Set is a Subset of every Set.
The Empty Set is also a Proper Subset of every Set except the Empty Set.
Number of Subsets – The number of distinct subsets of a set containing n
elements is given by 𝟐𝒏 .
Number of Proper Subsets – The number of distinct proper subsets of a set
containing n elements is given by 𝟐𝒏 − 𝟏.
Example: How many Subsets and Proper Subsets does Set A have?
𝑆𝑒𝑡 𝐴 = {𝑏𝑎𝑛𝑎𝑛𝑎𝑠, 𝑜𝑟𝑎𝑛𝑔𝑒𝑠, 𝑠𝑡𝑟𝑎𝑤𝑏𝑒𝑟𝑟𝑖𝑒𝑠}
𝑛=3
Subsets = 2 = 2 = 8 Proper Subsets = 2 − 1 = 7
Example: List the Proper Subsets for the Example above.
1. {𝑏𝑎𝑛𝑎𝑛𝑎𝑠} 5. {𝑏𝑎𝑛𝑎𝑛𝑎𝑠, 𝑠𝑡𝑟𝑎𝑤𝑏𝑒𝑟𝑟𝑖𝑒𝑠}
2. {𝑜𝑟𝑎𝑛𝑔𝑒𝑠} 6. {𝑜𝑟𝑎𝑛𝑔𝑒𝑠, 𝑠𝑡𝑟𝑎𝑤𝑏𝑒𝑟𝑟𝑖𝑒𝑠}
3. {𝑠𝑡𝑟𝑎𝑤𝑏𝑒𝑟𝑟𝑖𝑒𝑠} 7. ∅
4. {𝑏𝑎𝑛𝑎𝑛𝑎𝑠, 𝑜𝑟𝑎𝑛𝑔𝑒𝑠}
Intersection of Sets – The Intersection of Sets A and B is the set of elements that
are in both A and B, i.e. what they have in common. It can be written as 𝑨 ∩ 𝑩.
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4. Union of Sets – The Union of Sets A and B is the set of elements that are
members of Set A, Set B, or both Sets. It can be written as 𝑨 ∪ 𝑩.
Example: Find the Intersection and the Union for the Sets A and B.
𝑆𝑒𝑡 𝐴 = {𝑅𝑒𝑑, 𝐵𝑙𝑢𝑒, 𝐺𝑟𝑒𝑒𝑛} Set A and B only
have 2 elements in
𝑆𝑒𝑡 𝐵 = {𝑌𝑒𝑙𝑙𝑜𝑤, 𝑂𝑟𝑎𝑛𝑔𝑒, 𝑅𝑒𝑑, 𝑃𝑢𝑟𝑝𝑙𝑒, 𝐺𝑟𝑒𝑒𝑛}
common.
Intersection: 𝑨 ∩ 𝑩 = {𝑅𝑒𝑑, 𝐺𝑟𝑒𝑒𝑛}
Union: 𝑨 ∪ 𝑩 = {𝑅𝑒𝑑, 𝐵𝑙𝑢𝑒, 𝐺𝑟𝑒𝑒𝑛, 𝑌𝑒𝑙𝑙𝑜𝑤, 𝑂𝑟𝑎𝑛𝑔𝑒, 𝑃𝑢𝑟𝑝𝑙𝑒}
List each distinct element only once, even
if it appears in both Set A and Set B.
Complement of a Set - The Complement of
Set A, written as A’ , is the set of all elements in the given Universal Set (U),
that are not in Set A.
Example: Let 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and 𝐴 = {1,3, 5, 7, 9}
Find 𝐴′ . Cross off everything in U that is also in A. What is
left over will be the elements that are in A’
𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
So, 𝐴 = {2, 4, 6, 8, 10}