2. Given the following four propositions:
(1) If $A=\left\{y \mid y=x^{2}+1, x \in \mathbf{N}_{+}\right\}, B=\left\{y \mid y=x^{2}-2 x+2\right.$, $\left.x \in \mathbf{N}_{+}\right\}$, then $A=B$;
(2) If $A$ and $B$ are both non-empty sets, and $A \cap B=\varnothing, M=\{A$'s subsets $\}, P=\{B$'s subsets $\}$, then $M \cap P=\{\varnothing\}$;
(3) If $A=B$, then $A \cap C=B \cap C$;
(4) If $A \cap C=B \cap C$, then $A=B$.

The number of correct propositions is ( ).
A. 0
B. 1
C. 2
D. 3
2. C.
(1) $y=x^{2}-2x+2=(x-1)^{2}+1$, at this point $x-1 \geqslant 0$, so $A \subset B$; (2) (3) correct; (4) counterexample can be given: $C=\varnothing, A=\{1\}, B=\{2\}$.