5. In the coordinate plane, points with both coordinates as integers are called integer points. We use $I$ to denote the set of all lines, $M$ to denote the set of lines passing through exactly one integer point, $N$ to denote the set of lines not passing through any integer points, and $P$ to denote the set of lines passing through infinitely many integer points. Then, the number of correct statements among the following expressions:
(1) $M \cup N \cup P=I$;
(3) $N \neq \Phi$;
(2) $M \neq \boldsymbol{\Phi}$;
(4) $P \neq \Phi$.
is
(A) 1
(B) 2
(C) 3
(D) 4
Answer ( )
(D)
5.【Analysis and Solution】The line $y=\frac{1}{2}$ does not pass through any integer points, so expression (2) is correct; the line $y=\sqrt{2} x$ has infinitely many, passing through exactly one integer point $(0,0)$, so expression (3) is correct; the line $y=x$ passes through multiple integer points, so expression (4) is correct.
We will prove that if $a x+b x+c=0$ passes through two integer points $\left(x_{1}, y_{1}\right) 、\left(x_{2}, y_{2}\right)$, then it passes through infinitely many integer points $\left(x_{2}+\right.$
$$
\begin{array}{l}
\left.k\left(x_{2}-x_{1}\right), y_{1}+k\left(y_{2}-y_{1}\right)\right), k \in Z . \\
\quad \text { In fact } a+\left[\left(x_{1}+k\left(x_{2}-x_{1}\right)\right]+b\left[y_{1}+k\left(y_{2}-y_{1}\right)\right]+c\right. \\
\quad=a x_{1}+b y_{1}+c+k\left(a x_{2}+b y_{2}+c\right)-k\left(a x_{1}+b y_{1}+c\right) \\
\quad=0 \text {, hence the answer is (D). }
\end{array}
$$