Computability and Logic

7.1. RECURSIVE RELATIONS 77
Proof:
(a), (b): These have already been proved.
(c): In the remaining items, we write simply x for x 1 , ..., x n . The characteristic
function c ∗ of the negation or complement of R is obtainable from the characteristic
function c of R by c ∗ (x) = 1 − . c(x).
(d), (e): The characteristic function c ∗ of the conjunction or intersection of R 1
and R 2 is obtainable from the characteristic functions c 1 and c 2 of R 1 and R 2 by
c ∗ (x) = min(c 1 (x), c 2 (x)), and the characteristic function c† of the disjunction or
union is similarly obtainable using max in place of min.
(f), (g): From the characteristic function c(x, y) of the relation R(x, y) the characteristic
functions u and e of the relations ∀v ≤ yR(x 1 , ..., x n , v) and ∃v ≤ yR(x 1 ,
..., x n , v) are obtainable as follows:
(
y∏
y∑
)
u(x, y) = c(x, i) e(x, y) = sg c(x, i)
i=0
i=0
where the summation (∑) and product (∏) notation is as in Proposition 6.5. For the
product will be 0 if any factor is 0, and will be 1 if and only if all factors are 1; while
the sum will be positive if any summand is positive. For the strict bounds ∀v < y and
∃v < y we need only replace y by y . − 1.
7.5 Example (Primality). Recall that a natural number x is prime if x > 1 and there do
not exist any u, v both