summing terms to the cardinality all possible subsets of a given set

Question

I am facing the problem of computing the sum $$\sum_{L\subset S}(-1)^{|S|-|L|},$$ where $|S|$ denotes the cardinality of a set $S$ is finite and $L$ is a proper subset of $S$. Thank you!

Answer 1

Given that the cardinallity of $\{L\subset S: |L| = j\}$ is $\binom{|S|}{j}$, we have: $$\sum_{L\subsetneq S}(-1)^{|S|-|L|} = \sum_{j=0}^{|S|-1}\sum_{L\subset S,\\|L|=j}(-1)^{|S|-|L|} = \\ = \sum_{j=0}^{|S| - 1}\binom{|S|}{j}(-1)^{|S|-j} $$

Does this look familiar?

Answer 2

Finding a subset of $|S|$ of cardinality $|L|$ is the same as finding a subset of cardinality $|S|-|L|$ so let us rewrite the sum as $$\sum_{L\subset S}(-1)^{|L|}.$$ Now, there is exactly ${|S|}\choose{|L|}$ subset of size $|L|$ in $S$. So the sum becomes $$\sum_{|L|=0}^{|S|-1}{{|S|}\choose{|L|}}(-1)^{|L|} = (-1)^{|S|}$$ using Newton's binomial formula.