Why Does the Power Set of a Set Have 2^n Elements?

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SUMMARY

The power set of a set B containing n elements has exactly 2^n subsets, as established in combinatorial mathematics. Each element of the set can either be included or excluded from a subset, leading to a binary representation of choices, which results in 2^n possible combinations. This principle is further illustrated through the binomial expansion of (1+1)^n, confirming the relationship between the number of subsets and the size of the original set.

PREREQUISITES
  • Understanding of set theory and subsets
  • Familiarity with combinatorial principles
  • Knowledge of binary representation
  • Basic grasp of binomial expansion
NEXT STEPS
  • Study the concept of binomial coefficients and their relation to subsets
  • Explore combinatorial proofs for the power set size
  • Learn about binary strings and their applications in set theory
  • Investigate advanced topics in combinatorics, such as the inclusion-exclusion principle
USEFUL FOR

Students of mathematics, particularly those studying combinatorics, educators teaching set theory, and anyone interested in the foundational concepts of mathematical logic and binary systems.

michonamona
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Homework Statement





Why is the size of the power set 2^n ?

To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ?

It boggles my mind why the base is 2 for all size of sets.

Thank you,

M

Homework Equations





The Attempt at a Solution

 
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michonamona said:

Homework Statement





Why is the size of the power set 2^n ?

To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ?

It boggles my mind why the base is 2 for all size of sets.

Thank you,

M

Homework Equations





The Attempt at a Solution

Haha, keep doing combinatorics and a lot of stuff will blow your mind.


I don't want to give too much away, here, I'll be around to help if you need it, but think about the set and the power set of that set as a binary string of length n where each element of the string represents an element of the set.
 
michonamona said:

Homework Statement





Why is the size of the power set 2^n ?

To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ?

It boggles my mind why the base is 2 for all size of sets.

Thank you,

M

Homework Equations





The Attempt at a Solution


If you have a set with n elements, now many subsets of size 0 are there? Of size 1? Size 2?...Size n? How many total then?

Then think about the binomial expansion of (1+1)n.
 

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