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Sets and their subsets

Contributed by:
NEO
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: 𝑆𝑒𝑑 𝐴 = {π‘Ž, 𝑏, 𝑐, 𝑑} π‘œπ‘Ÿ 𝑆𝑒𝑑 𝐡 = {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 β€œRoster Method.” A
description can also be used in the braces. This is called β€œSet-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.: {π‘₯ π‘₯ 𝑖𝑠 π‘Ž π‘›π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™ π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘“π‘Ÿπ‘œπ‘š 1 π‘‘π‘œ 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: {π‘Ž, 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} , 𝐡 = {4, 3, 2, 1, 2, 3, 4, } .
Sets 𝐴 π‘Žπ‘›π‘‘ 𝐡 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 = {πΉπ‘œπ‘œπ‘‘π‘π‘Žπ‘™π‘™, π΅π‘Žπ‘ π‘˜π‘’π‘‘π‘π‘Žπ‘™π‘™, π΅π‘Žπ‘ π‘’π‘π‘Žπ‘™π‘™, π‘†π‘œπ‘π‘π‘’π‘Ÿ} Both Sets have 4
𝐡 = {𝑝𝑒𝑛𝑛𝑦, π‘›π‘–π‘π‘˜π‘’π‘™, π‘‘π‘–π‘šπ‘’, π‘žπ‘’π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿ} elements
𝑛(𝐴) = 4 π‘Žπ‘›π‘‘ 𝑛(𝐡) = 4
𝐴 π‘Žπ‘›π‘‘ 𝐡 π‘Žπ‘Ÿπ‘’ πΈπ‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ 𝑆𝑒𝑑𝑠, π‘šπ‘’π‘Žπ‘›π‘–π‘›π‘” 𝑛(𝐴) = 𝑛(𝐡).
Note: If two sets are Equal, they are also Equivalent!
Example: 𝑆𝑒𝑑 𝐴 = {π‘Ž, 𝑏, 𝑐, 𝑑} 𝑆𝑒𝑑 𝐡 = {𝑑, 𝑑, 𝑐, 𝑐, 𝑏, 𝑏, π‘Ž, π‘Ž}
Are Sets A and B Equal? Sets A and B have the οƒ  Yes!
exact same elements!
{π‘Ž, 𝑏, 𝑐, 𝑑}
Are Sets A and B Sets A and B have the
Equivalent? exact same number of οƒ  Yes!
distinct elements!
𝑛(𝐴) = 𝑛(𝐡) = 4
The Empty Set (or Null Set) – The set that contains no elements.
It can be represented by either { } π‘œπ‘Ÿ βˆ….
Note: Writing the empty set as {βˆ…} is not correct!
Symbols commonly used with Sets –
∈ β†’ π‘–π‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘  π‘Žπ‘› π‘œπ‘π‘—π‘’π‘π‘‘ 𝑖𝑠 π‘Žπ‘› π’†π’π’†π’Žπ’†π’π‘‘ π‘œπ‘“ π‘Ž 𝑠𝑒𝑑.
∈ β†’ π‘–π‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘  π‘Žπ‘› π‘œπ‘π‘—π‘’π‘π‘‘ 𝑖𝑠 𝒏𝒐𝒕 π‘Žπ‘› π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘œπ‘“ π‘Ž 𝑠𝑒𝑑.
 β†’ π‘–π‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘  π‘Ž 𝑠𝑒𝑑 𝑖𝑠 π‘Ž 𝒔𝒖𝒃𝒔𝒆𝒕 π‘œπ‘“ π‘Žπ‘›π‘œπ‘‘β„Žπ‘’π‘Ÿ 𝑠𝑒𝑑.
οƒŒ β†’ π‘–π‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘  π‘Ž 𝑠𝑒𝑑 𝑖𝑠 π‘Ž 𝒑𝒓𝒐𝒑𝒆𝒓 𝒔𝒖𝒃𝒔𝒆𝒕 π‘œπ‘“ π‘Žπ‘›π‘œπ‘‘β„Žπ‘’π‘Ÿ 𝑠𝑒𝑑.
∩ β†’ π‘–π‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘  π‘‘β„Žπ‘’ π’Šπ’π’•π’†π’“π’”π’†π’„π’•π’Šπ’π’ π‘œπ‘“ 𝑠𝑒𝑑𝑠.
βˆͺ β†’ π‘–π‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘  π‘‘β„Žπ‘’ π’–π’π’Šπ’π’ π‘œπ‘“ 𝑠𝑒𝑑𝑠.
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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 β‰  𝑩)
It is written as 𝑨 οƒŒ 𝑩.
Example: 𝑆𝑒𝑑 𝐴 = {2, 4, 6} 𝑆𝑒𝑑 𝐡 = {0, 2, 4, 6, 8}
{2, 4, 6}  {0, 2, 4, 6, 8} and {2, 4, 6} οƒŒ {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 β‰  𝐡. 𝑨 οƒŒ 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 πŸπ’ .
Number of Proper Subsets – The number of distinct proper subsets of a set
containing n elements is given by πŸπ’ βˆ’ 𝟏.
Example: How many Subsets and Proper Subsets does Set A have?
𝑆𝑒𝑑 𝐴 = {π‘π‘Žπ‘›π‘Žπ‘›π‘Žπ‘ , π‘œπ‘Ÿπ‘Žπ‘›π‘”π‘’π‘ , π‘ π‘‘π‘Ÿπ‘Žπ‘€π‘π‘’π‘Ÿπ‘Ÿπ‘–π‘’π‘ }
𝑛=3
Subsets = 2 = 2 = 8 Proper Subsets = 2 βˆ’ 1 = 7
Example: List the Proper Subsets for the Example above.
1. {π‘π‘Žπ‘›π‘Žπ‘›π‘Žπ‘ } 5. {π‘π‘Žπ‘›π‘Žπ‘›π‘Žπ‘ , π‘ π‘‘π‘Ÿπ‘Žπ‘€π‘π‘’π‘Ÿπ‘Ÿπ‘–π‘’π‘ }
2. {π‘œπ‘Ÿπ‘Žπ‘›π‘”π‘’π‘ } 6. {π‘œπ‘Ÿπ‘Žπ‘›π‘”π‘’π‘ , π‘ π‘‘π‘Ÿπ‘Žπ‘€π‘π‘’π‘Ÿπ‘Ÿπ‘–π‘’π‘ }
3. {π‘ π‘‘π‘Ÿπ‘Žπ‘€π‘π‘’π‘Ÿπ‘Ÿπ‘–π‘’π‘ } 7. βˆ…
4. {π‘π‘Žπ‘›π‘Žπ‘›π‘Žπ‘ , π‘œπ‘Ÿπ‘Žπ‘›π‘”π‘’π‘ }
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 βˆͺ 𝑩.
Example: Find the Intersection and the Union for the Sets A and B.
𝑆𝑒𝑑 𝐴 = {𝑅𝑒𝑑, 𝐡𝑙𝑒𝑒, πΊπ‘Ÿπ‘’π‘’π‘›} Set A and B only
have 2 elements in
𝑆𝑒𝑑 𝐡 = {π‘Œπ‘’π‘™π‘™π‘œπ‘€, π‘‚π‘Ÿπ‘Žπ‘›π‘”π‘’, 𝑅𝑒𝑑, π‘ƒπ‘’π‘Ÿπ‘π‘™π‘’, πΊπ‘Ÿπ‘’π‘’π‘›}
common.
Intersection: 𝑨 ∩ 𝑩 = {𝑅𝑒𝑑, πΊπ‘Ÿπ‘’π‘’π‘›}
Union: 𝑨 βˆͺ 𝑩 = {𝑅𝑒𝑑, 𝐡𝑙𝑒𝑒, πΊπ‘Ÿπ‘’π‘’π‘›, π‘Œπ‘’π‘™π‘™π‘œπ‘€, π‘‚π‘Ÿπ‘Žπ‘›π‘”π‘’, π‘ƒπ‘’π‘Ÿπ‘π‘™π‘’}
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 π‘ˆ = {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’
π‘ˆ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
So, 𝐴 = {2, 4, 6, 8, 10}
Try these on your own!
Given the set descriptions below, answer the following questions.
π‘ˆ = 𝐴𝑙𝑙 πΌπ‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘“π‘Ÿπ‘œπ‘š 1 π‘‘π‘œ 10. 𝐴 = 𝑂𝑑𝑑 πΌπ‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘“π‘Ÿπ‘œπ‘š 1 π‘‘π‘œ 10,
𝐡 = 𝐸𝑣𝑒𝑛 πΌπ‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘“π‘Ÿπ‘œπ‘š 1 π‘‘π‘œ 10, 𝐢 = 𝑀𝑒𝑙𝑑𝑖𝑝𝑙𝑒𝑠 π‘œπ‘“ 2 π‘“π‘Ÿπ‘œπ‘š 1 π‘‘π‘œ 10.
1. Write each of the sets in roster notation. π‘ˆ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} , 𝐴 = {1, 3, 5, 7, 9} ,
𝐡 = {2, 4, 6, 8, 10} , 𝐢 = {2, 4, 6, 8, 10}
2. What is the cardinality of Sets U and A? Cardinality: U 10, A  5
3. Are Set B and Set C Equal? Yes, they are Equal
4. Are Set A and Set C Equivalent? Yes, they are Equivalent
5. How many Proper Subsets of Set π‘ˆare there? 2 βˆ’ 1 = 1023
6. Find 𝑩 π‘Žπ‘›π‘‘ π‘ͺβ€² 𝐡 = 𝐢 = {1, 3, 5, 7, 9}
7. Find 𝑨 βˆͺ π‘ͺβ€² 𝐴 βˆͺ 𝐢 = {1, 3, 5, 7, 9}
8. Find 𝑩 ∩ π‘ͺ 𝐡 ∩ 𝐢 = { } or βˆ…
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Mathematics Mathematics - Grade 7 - United States Number Theory