propositional functions and set operators

• Feb 17th 2013, 10:17 PM
eaksoy
propositional functions and set operators
Hi guys,
I'm new on forum. With searching I got some answers of my questions. TY for that. But I couldn't find exact answer for this Q's.

1) Are all functions are propositional functions or not?
2) How can we relate union, intersection, complement with logical connectives (AND, OR, NOT)?

Thanks
• Feb 18th 2013, 06:28 AM
emakarov
Re: propositional functions and set operators
Quote:

Originally Posted by eaksoy
1) Are all functions are propositional functions or not?

No, functions from real numbers to real numbers are not propositional just because their domain and codomain (range) are not the set {T, F} of truth values.

Quote:

Originally Posted by eaksoy
2) How can we relate union, intersection, complement with logical connectives (AND, OR, NOT)?

The connection is made using characteristic functions. Given a subset S of some universal set U, the characteristic function of S, $\displaystyle \chi_S:U\to\{T,F\}$, is defined as follows.

$\displaystyle \chi_S(x)=\begin{cases}T & x\in S\\F & x\notin S\end{cases}$

(Usually 1 and 0 are used instead of T and F.) Then for any sets A, B, we have $\displaystyle \chi_{A\cap B}(x)=\chi_A(x)\text{ AND }\chi_B(x)$, and similarly for other set operations.