Power Set – Set Theory day 2
I.
Set A is said to be a subset of B iff for every x  x A  x B 
(Element x is contained in set A implies Element x is contained in set B).
A B
means A is a subset of B
Example:
F= {1, 2, 3, 4, 5}
B = {2, 4}
C = {6}
You can see that all of set B is in set F so we can conclude: B  F
But since C does not have anything in F or B it is not a subset of either: C  F
Is the empty set a subset of F?
What are all the subset of C if
yes, the empty set is a subset of every set.
Set C = {1, 2}
P  C   {,{1},{2},{1, 2}}
What are all the subsets of A if Set A = {1, 2, 3}
P  A  {,{1},{2},{3},{1, 2},{1,3},{2,3},{1, 2,3}}
What is the subset of B is Set B = {1, 2, 3, 4}
P  B   {,{1},{2},{3},{4},{1, 2},{1,3},{1, 4},{2,3},
{2, 4},{3, 4},{1, 2,3},{1, 2, 4},{1,3, 4},{2,3, 4},{1, 2,3, 4}}
The Cardinal number of each set is the number of elements in the set.
So from above, what is the Cardinality of each of the sets?
Set A’s cardinality is 8.
Set B’s cardinality is 16.
Set C’s cardinality is 4.
P    {} which is 1 element.
P {a}  {,{a}} which has 2 elements
P {a, b}  {,{a},{b},{a, b}} which has 4 elements
Do you see a pattern?
P {a, b}  {,{a},{b},{a, b}}
P    {} which is 20  1 element.
P {a}  {,{a}} which has 21  2 elements
P {a, b}  {,{a},{b},{a, b}} which has 22  4 elements
So we can conclude that a power set has n elements so
P  n # elements   2n elements