a)      ~(p ^ q) = ~(F ^ T) =  ~(F) = T
b)	 p v ~q  =   F v ~T = F v F = F 
c)      ~(p → q) = ~(F → T) =  ~(F) = T
d)      ~p → q   =   ~F → T = T → T = T
e)	p ↔ ~q   =   F ↔ ~T = F ↔ F = T  

~ is T when followed by F and vice-versa.
^ is T only when it has a T on each side of it, otherwise it's F
v is T unless it has 2 F's on each side of it.
→ is T unless it is a case of T→F 

p  q  ~p  ~p v q |  p → q
-----------------|-------
T  T   F     T   |    T      
T  F   F     F   |    F 
F  T   T     T   |    T
F  F   T     T   |    T

They are equivalent because both their truth tables go TFTT.

Edwin