0

We got to show the following equality:

$1_{A_1 \cup...\cup A_n} = 1 - \prod \limits_{i=1}^n (1 - 1_{A_i})$

First I would like to ask for hints for how to proove this equation (no solution though, I would like to solve it myself). Secondly I was wondering if there is an "intuitive" interpretation of this equality?

user62487
  • 483

1 Answers1

2

Yes. De Morgan's law for Boolean algebras:

$A\cup B=(A^c\cap B^c)^c$

where $X^c$ denotes the complement set. That's the hint.

Berci
  • 90,745
  • When I understand it correctly then, the above equality can be shown as follows to hold: Either $x \in A_1 \cup .. \cup A_n$ or $x \notin A_1 \cup ... \cup A_n$. In accordance with the left side the right side then will be either 1 in the first case or 0 in the last case. – user62487 Feb 26 '13 at 16:38
  • The complement corresponds to the unary operation $f\mapsto 1-f$ and the intersection corresponds to multiplication among characteristic functions. – Berci Feb 26 '13 at 20:44
  • Thanks for this helpful remark. I understand :-) – user62487 Feb 26 '13 at 21:15