How many elements are in two power sets

[chocolatelover]

[SOLVED] how many elements are in two power sets

Homework Statement

a.How many elements are in the power set {1,2,3,4}? b.How many elements are in the power set {1,2,3,4,5}?

Homework Equations

The Attempt at a Solution

a. {empty set, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,3,4}} Are there 13 elements?

b. {empty set, {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,3,4}}

Are there 19 elements?

Is there a general rule as to how many elements are in a power set?

Thank you very much

 

[Shooting Star]

The Attempt at a Solution

a. {empty set, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,3,4}} Are there 13 elements?

What happened to {2,3,4} and a few more?

b. {empty set, {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,3,4}}

Are there 19 elements?

No. Like the last one, some are mising.

Is there a general rule as to how many elements are in a power set?

It will be just the total number of combinations of 'n' things taken any number at a time, and one more (when you don't select any). Do you know how to find that?

 

[chocolatelover]

I got 16 elements in the first one. Does that look right?

(It will be just the total number of combinations of 'n' things taken any number at a time, and one more (when you don't select any). Do you know how to find that?)

No, I don't know how to do that. Could you please show me?

Thank you

 

[Shooting Star]

I got 16 elements in the first one. Does that look right?

That is correct.

Consider the first problem. You can choose 1, or not choose 1. So, 1 can be dealt with in two ways. For each of these cases, you can choose, or not choose 2. So, total number of ways to select either 1 or 2 or both or none is 2X2=4=2^2. Choosing none means you get the null set. Similarly, for the four elements 1 2 3 4, total number of choosing would be 2^4. All these selections is giving you the elements of the power set. So, the number of elements in the power set of the set {1,2,3,4} is 2^4.

I'm sure you can tackle the next problem using this formula for the number of elements in the power set of a finite set.

 

[chocolatelover]

Thank you very much

Regards

 

[sutupidmath]

Just writitng in general what actually Shooting Star already said and elaborated.

If X is a set with |X|=n elements then: $$|P(X)|=2^{|X|}=2^{n}$$ so the power set has 2^n elements.

 

[chocolatelover]

Thank you very much

Regards