Respuesta :
Answer:
number of subsets of a set with 13 elements are: [tex]2^{13}[/tex]
Step-by-step explanation:
In order to solve this intuitively, we can start by a set with lesser elements. This will reveal a pattern that will be used to solve for the subsets of the 13 element set.
If we start with a set B. which contains only 3 elements.
[tex]B = \{1,2,3\}[/tex]
how many subsets of B are there? well we can count them. [the set containing {1,2} and {2,1} are the same, arrangement doesn't matter]
[tex]B_{0} = \{\}\\B_{1a}=\{1\}\\B_{1b}=\{2\}\\B_{1c}=\{3\}\\B_{2a}=\{1,2\}\\B_{2b}=\{2,3\}\\B_{2c}=\{3,1\}\\B_{3a}=\{1,2,3\}\\[/tex]
there are a total of 9 subsets here.
Similarly, if you try a with a subset with only two elements you'll find that it has a total of 4 subsets.
We can see that combinatorics is at play here.
for the set B. the number of subsets can be written as:
[tex]\text{\# of subsets of B} = ^3C_0+^3C_1+^3C_2+^3C_3\\\text{\# of subsets of B} = 1+3+3+1\\\\text{\# of subsets of B} = 8[/tex]
if we try with a 2-element set:
[tex]\text{\# of subsets} = ^2C_0+^2C_1+^2C_2\\\text{\# of subsets} = 1+2+1\\\ \text{\# of subsets} = 4[/tex]
We can use the same technique to find the number of subsets of the 13 element set.
But if you recognize a pattern here that this sets of combinations are actually part of the pascal triangle, the sum of each row of the triangle is 2^{the row's number}. hence.
[tex]\text{\# of subsets of B} = 2^3\\\ \text{\# of subsets of B} = 8[/tex]
So finally, the subsets of a 13-element set A will be
[tex]\text{\# of subsets of A} = ^{13}C_0+^{13}C_1+^{13}C_2+^{13}C_3\cdots+^{13}C_{12}+^{13}C_{13}\\OR\\\text{\# of subsets of A} = 2^{13}\\\text{\# of subsets of A} = 8192[/tex]
If the set A has 13 elements, the number of different subsets is [tex]2^{13}=8192[/tex]
All the possible subsets that can be formed from any given set is called the Power set of that set. Generally, if we had a set [tex]H[/tex] such that
[tex]|H|=k[/tex]
Where [tex]|H|[/tex] denotes the cardinality, or number of elements, in [tex]H[/tex], the power set of [tex]H[/tex], denoted by [tex]P(H)[/tex], has the following formula
[tex]P(H)=2^k\text{ elements}[/tex]
So, given the set [tex]A[/tex] such that
[tex]|A|=13[/tex]
the power set of [tex]A[/tex] will have [tex]2^{13} \text{ or } 8192 \text{ elements}[/tex]
Learn more about number of different subsets here: https://brainly.com/question/13266391
