This is my homework and the problem is to translate below into predicated logic and prove it with different methods:
Determine if the following argument is a valid inference.“all the mammals are vertebrates; Not all of the mammals are viviparous animals; So, some vertebrates are not viviparous.”
My only solution is:
$P(x): x \text{ is a mammal}$
$Q(x): x \text{ is a vertebrate}$
$R(x): x \text{ is viviparous}$
Premises: $\forall x (P(x) \Rightarrow Q(x)), \exists x (P(x) \wedge \sim R(x))$
Conclusion: $\exists x (Q(x) \wedge \sim R(x))$
$$ \begin{align} &(1) \exists x (P(x) \land \sim R(x)) \tag{premise} \\ &(2)P(a) \land \sim R(a) \tag{ES} \\ &(3) \forall x (P(x) \Rightarrow Q(x)) \tag{premise} \\ &(4)P(a) \Rightarrow Q(a) \tag{US} \\ &(5) P(a) \tag{simplification of (2)} \\ &(6) \sim R(a) \tag{simplification of (2)} \\ &(7)Q(a) \tag{modus ponens} \\ &(8)Q(a) \land \sim R(a) \tag{conjunction of (6), (7)} \\ &(9) \exists x (Q(x) \land \sim R(x)) \tag{EG} \end{align} $$
Is there any other way to give the proof? Thanks a lot!