Taking Things Apart
Are you fond of taking things apart?
OBJECTIVE: To learn about subsets
In previous lessons, you learned about sets and some of their properties.
Setting the Table: https://www.geogebra.org/m/eqajxyqr
Counting with Your Fingers: https://www.geogebra.org/m/kmqughqr
In this lesson, you're going to learn about subsets and more set vocabulary.
SET VOCABULARY
The sets below will be used to cite examples for the new set vocabulary.
Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3}
B = {1, 2, 3, 4}
C = {1, 3, 5}
D = {4, 6, 8}
E = {2, 4, 6, 8}
F = {2x | x is a positive integer less than 5}
Universal Set—the totality of sets under a particular discussion, denoted as U
Example: U = {positive integers greater than or equal to 1 and less than or equal to 9}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Subset—set all of whose elements are contained in another set
Proper Subset—subset of another set, but excluding at least one element of that set
Improper Subset—subset of another set, but including all the elements of that set
⊂—"is a proper subset of"
⊄—"is not a proper subset of"
⊆—"is an improper subset of"
Examples: A ⊂ B, C ⊄ D, E ⊆ F
Superset—set that contains all the elements of another set
Proper Superset—superset of another set, but including at least one element more than that set
Improper Superset—superset of another set, but including all the elements of that set and nothing more
⊃—"is a proper superset of"
⊅—"is not a proper superset of"
⊇—"is an improper superset of"
Examples: B ⊃ A, D ⊅ C, F ⊇E
Power Set—set containing all the subsets of another set, denoted as P(X)
Example: P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
NOTES: (1) The empty set (∅) is a subset of every set.
(2) Any set is a subset of itself.
(3) The cardinality of the Power Set is given by the formula 2ⁿ, where n is the cardinality of the set.
In the example above, n(A) = 3. Therefore, nP(A) = 2³ or 8.
Joint Sets—two sets with at least one element in common
Examples: A and B are joint sets. A and C are also joint sets.
Disjoint Sets—two sets with no element in common
Examples: A and D are disjoint sets. C and E are also disjoint sets.
Complement of a Set—set whose elements are not in that set, but are in the universal set, denoted as X' or Xᶜ.
Examples: A' = {4, 5, 6, 7, 8, 9}, Eᶜ = {1, 3, 5, 7, 9}
Below is a set of problems involving sets and subsets.
Sets and Subsets
ANSWER BOX:
Check the Answer Box below for the correct answers.
In this lesson, you learned about subsets and their relationship to sets.
In succeeding lessons, you'll learn about operations on sets. Did you ENJOY today's lesson?