1. Section 2.2 Subsets

2. What You Will Learn
Subsets and proper subsets

3. Subsets
Set A is a subset of set B, symbolized A ⊆ B, if and only if all elements of set A are also elements of set B.
The symbol A ⊆ B indicates that “set A is a subset of set B.”

4. Subsets The symbol A ⊈ B set A is not a subset of set B.
To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B.

5. Determining Subsets Example:
Determine whether set A is a subset of set B.
A = { 3, 5, 6, 8 }
B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Solution:
All of the elements of set A are contained in set B, so A ⊆ B.

6. Proper Subset
Set A is a proper subset of set B, symbolized A ⊂ B, if and only if all of the elements of set A are elements of set B and set A ≠ B (that is, set B must contain at least one element not is set A).

7. Determining Proper Subsets
Example:
Determine whether set A is a proper subset of set B.
A = { dog, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B, and sets A and B are not equal, therefore A ⊂ B.

8. Determining Proper Subsets
Example:
Determine whether set A is a proper subset of set B.
A = { dog, bird, fish, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B, but sets A and B are equal, therefore A ⊄ B.

9. Number of Distinct Subsets
The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A.

10. Number of Distinct Subsets
Example:
Determine the number of distinct subsets for the given set {t, a, p, e}.
List all the distinct subsets for the given set {t, a, p, e}.

11. Number of Distinct Subsets
Solution:
Since there are 4 elements in the given set, the number of distinct subsets is
24 = 2 • 2 • 2 • 2 = 16.
{t,a,p,e}, {t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { }

12. Number of Distinct Proper Subsets
The number of distinct proper subsets of a finite set A is 2n – 1, where n is the number of elements in set A.

13. Number of Distinct Proper Subsets
Example:
Determine the number of distinct proper subsets for the given set {t, a, p, e}.

14. Number of Distinct Subsets
Solution:
The number of distinct proper subsets is
24 – 1= 2 • 2 • 2 • 2 – 1 = 15.
They are {t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { }.
Only {t,a,p,e}, is not a proper subset.