Combinatorics and sets 2

Homework Statement

Given {1,2,...,n}, n is an even number. What are all the possible subsets that contain only even numbers? (Notice that ∅ is also defined as such a subset).

Homework Equations

2n - all possibilities for group A with n objects

The Attempt at a Solution

I think the answer is 2n/2 .... to include just even numbers. Is my reasoning correct?

FactChecker
Gold Member
I think the answer is 2n/2 .... to include just even numbers. Is my reasoning correct?
You should add a few sentences to explain your reasoning. And the question is "what are", not "how many", so can you say something more about them?.

You should add a few sentences to explain your reasoning. And the question is "what are", not "how many", so can you say something more about them?.

Just bad translation on my part. "How many" is more correct. As for reasoning, we were taught in class that for n objects, you have 2n possibilities for subsets. So I thought that because we have n/2 objects (we want only even numbers) 2n/2 is the answer.

FactChecker
Gold Member
Just bad translation on my part. "How many" is more correct. As for reasoning, we were taught in class that for n objects, you have 2n possibilities for subsets. So I thought that because we have n/2 objects (we want only even numbers) 2n/2 is the answer.
That is better. It never hurts to add a brief explanation like that.

That is better. It never hurts to add a brief explanation like that.

But is it correct?

PeroK
Homework Helper
Gold Member
2020 Award
But is it correct?
It is correct. The odd numbers are irrelevant.

It is correct. The odd numbers are irrelevant.

Hmm ok. But what about ∅ ? If it is even, it should be included, but it does not seem to be included in {1...n}...

PeroK
Homework Helper
Gold Member
2020 Award
That's the empty set. That is included as one of the ##2^n## subsets. If you are looking for non-empty subsets then there are only ##2^n -1## of those.

In this case you were explicitly told to count the empty set.

That's the empty set. That is included as one of the ##2^n## subsets. If you are looking for non-empty subsets then there are only ##2^n -1## of those.

In this case you were explicitly told to count the empty set.
Oh now I got it. Thanks!