85. Many definitions we have seen throughout the course can be stated using quantifiers.
For instance, to say that the integer $n$ is odd means $n$ has the following property:
$\exists k \in \mathbb{Z}, n = 2k + 1$
Answer each of the following using a specific term we have seen in class. (Don't
just convert the expression directly as written into an English sentence.)
a. What does the following statement mean about the integer $n$? (The answer will
involve "mod".)
$\exists q \in \mathbb{Z}, n = 5q + 3$
b. Suppose $n$ is an integer $\ge 2$. What does the following statement mean about $n$?
$\forall r, s \in \mathbb{Z}^+, (n = rs) \implies ((r = 1) \lor (s = 1))$
c. What does the following statement mean about the real number $x$?
$\exists a, b \in \mathbb{Z}, (b \ne 0) \land (x = \frac{a}{b})$
d. Suppose $f: X \to Y$ is a function. What does the following statement mean about $f$?
$\forall y \in Y, \exists x \in X, f(x) = y$
