NUMBER OF SUBSETS
If every element of a set B is also a member of a set A, then we say B is a subset of A. We use the symbol ⊂ to mean “is a subset of” and the symbol ⊄ to mean “is not a subset of”.
Example:
A = {1, 3, 5}, B = {1, 2, 3, 4, 5}
So, A ⊂ B because every element in A is also in B.
X = {1, 3, 5}, Y = {2, 3, 4, 5, 6}.
X ⊄ Y because 1 is in X but not in Y.
Note:
Example:
List all the subsets of the set Q = {x, y, z}
Solution:
The subsets of Q are { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}and {x, y, z}
a set of elements is given by the
For sets of , 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ...
A set is a collection of objects.
A set is definite if all its members are known. There are a lot of sets: a set of letters, a set of numbers....
letters = {a, b, c, ...x, y z}
numbers = {0, 1, 2, 3, ...}
Three points ... have the same meaning as etc. We use the symbol = and brackets {,} to denote the fact that specific elements belong to a set.
We denote sets with uppercase letters, for example: A, B, C, S, X, Y,…
The elements of sets are denoted by lowercase letters: a, b, c, x, y,….
The relation that an element x belongs to the set S is denoted by x S (we read it: “x is an element of S”).
If the element y is not a member of the set S, we write y S (we read it: “y is not an element of S”).
Find the number of subsets of the set.
{mom, dad, son, daughter}
a. 8
b. 16
c. 14
d. 12
Solution:
Well let's see if we can brute force it
1. { } empty set is a subset of every set
2. {mom, dad, son, daughter}
3. {mom}
4. {dad}
5. {son}
6. {daughter}
7. {mom, dad}
8. {mom, son}
9. {mom, daughter}
10. {dad, son}
11. {dad, daughter}
12. {son, daughter}
13. {mom, dad, son}
14, {mom, dad, daughter}
15. {mom, son, daughter}
16. {dad, son, daughter}
And that's it. 16 subsets.
Another way to determine this is 2^n where n is the number of elements in the set.
For this problem we had 4 elements so we have 2^4 subsets and 2^4 is 16.