$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$

Question

I have found this equality: $$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$$ Do you think is it true?

Answer 1

Yes this is true. You can think of this combinatorially. For instance suppose we have $$(1+x)(1+y)$$

First we choose to multiply nothing together. That is we choose just the 1's. To get the first term 1. Then we choose to multiply just one term. Here we get $x$ and $y$. Then we choose to multiply two together to get $xy$. Thus we have $1 + x + y + xy$. These correspond to the sets $$\emptyset, \{x\}, \{y\}, \{x,y\}$$

Now we see this works for the case where $|I| = 2$. To prove this by induction we have, assume this holds for $|I| = N$. Now let $I' = I \cup \{x_{N+1}\}$.

$$(1+x_{N+1})(1+x_N)\cdots(1+x_1) = (1+x_N)\cdots(1+x_1) + x_{N+1} (1+x_N)\cdots(1+x_1)$$

Then applying our assumption:

$$=\sum_{J\subset I} \prod_{j\in J} x_j + x_{N+1} \sum_{J\subset I} \prod_{j\in J} x_j$$

$$=\sum_{J\subset I} \prod_{j\in J} x_j + \sum_{J\cup\{x_{N+1}\}\subset I} \prod_{j\in J\cup\{x_{N+1}\}} x_j$$

$$= \sum_{J\subset I'} \prod_{j\in J} x_j $$