Exercise in propositional logic. Which of the following arguments is valid? 
A. If it rains, then the grass grows. The worms are not happy unless it rains. Therefore, If the worms are happy , then the grass grows.
B. If the wind howls, then the wolf howls. If the wind howls, then the birds sing. Therefore, if the birds sing, then the wolf howls.
C. If the sun shines, then it is day. If the stars shine, then the sun does not shine. Therefore, if the stars shine, it is not day
D. Both A and C.

A. $(U \wedge (\neg V \Rightarrow \neg U) \wedge (V \Rightarrow W)) \Rightarrow W$
True. If the worms are happy, then it rains, then the grass grows.
B. $((U \Rightarrow V) \wedge (U \Rightarrow W)) \Rightarrow (V \Rightarrow W)$
Wrong. If the birds sing, you don't know anything else. Especially, you don't know if the wind howls, only the converse is true.
C. $((U \Rightarrow V) \wedge (W \Rightarrow \neg U)) \Rightarrow (W \Rightarrow \neg U)$
Wrong. If the stars shine, then the sun does not shine, but you can't conclude the sun does not shine. Since $U \Rightarrow V$ does not imply $\neg U \Rightarrow \neg V$, but $\neg V \Rightarrow \neg U$.
