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

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.