I'm looking for a function that will replicate logical operators. In particular consider the AND operator. Then I am looking for a function z = f(x,y) such that $$x,y,z \in \{0,1\}$$
f(0,0)=0 f(1,0)=0 f(0,1)=0 f(1,1)=1
More importantly I need to generalize over a space where Xi,Yi ∈ {0,1} and I need to do this for other sorts of operators too. (X XOR Y AND Z for example.)
So given a vector Xi and a vector Yi I need to be able to generate Zi w
If $x, y \in [0, 1]$ you can take $f(x, y) = xy$. This comes from replacing "true" and "false" with probabilities and from assuming that $x, y$ are probabilities of independent events. Similarly for NOT take $f(x) = 1 - x$, and for OR take $f(x, y) = 1 - (1 - x)(1 - y)$.
If this isn't what you're looking for then you're going to have to be more precise about what you want. If you want to do something with vectors with entries in $\{ 0, 1 \}$, what's wrong with applying the operations pointwise?
In one of the standard approaches to fuzzy logic--used in household appliances everywhere!--the truth values are taken to be real numbers in the unit interval, where $0$ represents false and $1$ represents true, and the Boolean connectives are defined by the following equations:
These values agree with the classical values when restricted to $0$ and $1$, and thus agree with Qiaochu's answer on the classical values, but they disagree on non-classical values. In particular, this version of fuzzy logic gives the same truth value to $p$ as $p\wedge p$, but this isn't the case with the probabilistic formulas, which (inappropriately) interprets $p\wedge p$ as a conjunction of independent events. Similarly, fuzzy logic also says $p$ is equivalent to $p\vee p$, whereas the probabilistic formula does not.
It is a fun exercise to show that fuzzy logic doesn't work with classical tautologies very well. For example, $p\vee \neg p$ is not uniformly value $1$, although it is always at least $\frac 12$. Indeed, a statement is a classical tuatology if and only if it gets fuzzy value at least $\frac 12$ on all input. There are numerous other such fun problems.
There are also numerous variations on the fuzzy theme, however, with many different fuzzy logics, and the Wikipedia page appears extensive.
There exists an uncountable family of functions which will work for AND here, and another uncountable family of functions for OR here. Any T-norm will work for AND here, as follows: For any T-norm T(a, 1)=a by "1" as the identity for T, and by commutation T(1, a)=a also. So, if "a" belongs to {0, 1} we have T(1, 1)=T(0, 1)=T(1, 0)=1. 0<=1. So, by monotonicity it follows that T(0, 0)<=T(0, 1)=0. So, T(0, 0)=0. Thus, every t-norm works out such that T(0, 0)=T(0, 1)=T(1, 0), T(1, 1)=1. Any S-norm (or T-conorm) will work here for OR, since S(a, 0)=a by 0 as the identity element for S. By commutation, we have S(0, a)=a. So, if "a" belongs to {0, 1}, S(0, 0)=0, S(1, 0)=1, S(1, 1)=1. 0<=1, so 1=S(0, 1)<=S(1, 1), and thus for any S-norm S(1, 1)=1. So, any S-norm will work here for OR. min for AND, max for OR, and 1-a for NOT generally seem to work best.
