1, premise: Ax(F(x)→G(x)), Ex(H(x)∧F(x))

Conclusion: Ex(H(x)∧G(x))

Prove:

( 1)Ex(H(x)∧F(x))

(2)H(c)∧F(c) ( 1)EI

(3)H(c) (2) Simplification

(4)F(c) (2) Simplification

(5) Introduce ax (f (x) → g (x))

(6)F(c)→G(c) (5)UI

(7)G(c) (4)(6) Hypothetical reasoning

(8)H(c)∧G(c) (3)(7) Conjunction

(9)Ex(H(x)∧G(x)) (8)EG

2、f(4)= & lt； 5，4 >，f(-3)= & lt； -2，3 & gt；

F is a single shot. Because if f(x)=f(y), then

F is not a surjection. Because |x|≥0, if the second element of the ordered pair in Z×Z is negative, there is no corresponding element in z.