Contributed by:
NEO
Tue, Mar 15, 2022 12:46 PM UTC
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
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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: πππ‘ π΄ = {π, π, π, π} ππ πππ‘ π΅ = {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 βRoster Method.β A description can also be used in the braces. This is called βSet-builderβ 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: π΄ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Set Builder Not.: {π₯ π₯ ππ π πππ‘π’πππ ππ’ππππ ππππ 1 π‘π 10} Ellipsis - Three dots (β¦) 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, π, π, β¦ , π§} Cardinality of a Set β The number of distinct elements in a set. Example: Set π΄: The days of the week Members of Set A: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday Cardinality of Set π΄ = π(π¨) = 7 Equal Sets β Two sets that contain exactly the same elements, regardless of the order listed or possible repetition of elements. Example: π΄ = {1, 1, 2, 3, 4} , π΅ = {4, 3, 2, 1, 2, 3, 4, } . Sets π΄ πππ π΅ are equal because they contain exactly the same elements (i.e. 1, 2, 3, & 4). This can be written as π¨ = π©. 1
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Equivalent Sets β Two sets that contain the same number of distinct elements. Example: π΄ = {πΉπππ‘ππππ, π΅ππ πππ‘ππππ, π΅ππ πππππ, ππππππ} Both Sets have 4 π΅ = {πππππ¦, ππππππ, ππππ, ππ’πππ‘ππ} elements π(π΄) = 4 πππ π(π΅) = 4 π΄ πππ π΅ πππ πΈππ’ππ£πππππ‘ πππ‘π , πππππππ π(π΄) = π(π΅). Note: If two sets are Equal, they are also Equivalent! Example: πππ‘ π΄ = {π, π, π, π} πππ‘ π΅ = {π, π, π, π, π, π, π, π} Are Sets A and B Equal? Sets A and B have the ο Yes! exact same elements! {π, π, π, π} Are Sets A and B Sets A and B have the Equivalent? exact same number of ο Yes! distinct elements! π(π΄) = π(π΅) = 4 The Empty Set (or Null Set) β The set that contains no elements. It can be represented by either { } ππ β
. Note: Writing the empty set as {β
} is not correct! Symbols commonly used with Sets β β β πππππππ‘ππ ππ ππππππ‘ ππ ππ πππππππ‘ ππ π π ππ‘. β β πππππππ‘ππ ππ ππππππ‘ ππ πππ ππ πππππππ‘ ππ π π ππ‘. ο β πππππππ‘ππ π π ππ‘ ππ π ππππππ ππ ππππ‘βππ π ππ‘. ο β πππππππ‘ππ π π ππ‘ ππ π ππππππ ππππππ ππ ππππ‘βππ π ππ‘. β© β πππππππ‘ππ π‘βπ ππππππππππππ ππ π ππ‘π . βͺ β πππππππ‘ππ π‘βπ πππππ ππ π ππ‘π . 2
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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 π¨ ο π©. 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. (π¨ β π©) It is written as π¨ ο π©. Example: πππ‘ π΄ = {2, 4, 6} πππ‘ π΅ = {0, 2, 4, 6, 8} {2, 4, 6} ο {0, 2, 4, 6, 8} and {2, 4, 6} ο {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. π¨ ο π© in B, but A β π΅. π¨ ο π© 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 β The number of distinct subsets of a set containing n elements is given by ππ . Number of Proper Subsets β The number of distinct proper subsets of a set containing n elements is given by ππ β π. Example: How many Subsets and Proper Subsets does Set A have? πππ‘ π΄ = {πππππππ , πππππππ , π π‘πππ€πππππππ } π=3 Subsets = 2 = 2 = 8 Proper Subsets = 2 β 1 = 7 Example: List the Proper Subsets for the Example above. 1. {πππππππ } 5. {πππππππ , π π‘πππ€πππππππ } 2. {πππππππ } 6. {πππππππ , π π‘πππ€πππππππ } 3. {π π‘πππ€πππππππ } 7. β
4. {πππππππ , πππππππ } Intersection of Sets β 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 π¨ β© π©. 3
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Union of Sets β 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 π¨ βͺ π©. Example: Find the Intersection and the Union for the Sets A and B. πππ‘ π΄ = {π
ππ, π΅ππ’π, πΊππππ} Set A and B only have 2 elements in πππ‘ π΅ = {ππππππ€, ππππππ, π
ππ, ππ’ππππ, πΊππππ} common. Intersection: π¨ β© π© = {π
ππ, πΊππππ} Union: π¨ βͺ π© = {π
ππ, π΅ππ’π, πΊππππ, ππππππ€, ππππππ, ππ’ππππ} 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 Aβ , is the set of all elements in the given Universal Set (U), that are not in Set A. Example: Let π = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and π΄ = {1,3, 5, 7, 9} Find π΄β² . Cross off everything in U that is also in A. What is left over will be the elements that are in Aβ π = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} So, π΄ = {2, 4, 6, 8, 10} Try these on your own! Given the set descriptions below, answer the following questions. π = π΄ππ πΌππ‘πππππ ππππ 1 π‘π 10. π΄ = πππ πΌππ‘πππππ ππππ 1 π‘π 10, π΅ = πΈπ£ππ πΌππ‘πππππ ππππ 1 π‘π 10, πΆ = ππ’ππ‘πππππ ππ 2 ππππ 1 π‘π 10. 1. Write each of the sets in roster notation. π = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} , π΄ = {1, 3, 5, 7, 9} , π΅ = {2, 4, 6, 8, 10} , πΆ = {2, 4, 6, 8, 10} 2. What is the cardinality of Sets U and A? Cardinality: Uο 10, A ο 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 πare there? 2 β 1 = 1023 6. Find π© πππ πͺβ² π΅ = πΆ = {1, 3, 5, 7, 9} 7. Find π¨ βͺ πͺβ² π΄ βͺ πΆ = {1, 3, 5, 7, 9} 8. Find π© β© πͺ π΅ β© πΆ = { } or β
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