Some of the following arguments are valid, and some are not. If valid, state by what rule of inference (universal modus ponens or tollens). If invalid, state whether by converse or inverse error.

1. All nerds are good at math.
   Buffy is good at math
     Buffy is a nerd.
   
   
   
2. It is difficult to study when I am tired.
   I found it easy to study today.
      I am not tired today.
   
   
   
3. x, k  Z, if x * k is negative, then either x is negative or k is negative (x  k).
   x * k is positive
      x and k are both positive, or both negative.
   
   
   
4. Animals at the bottom of the food chain are very nervous.
   People are not at the bottom of the food chain.
      People are not nervous.
   
   
   
5. I'm always in a good mood when the weather is good.
   The weather today is great.
     I'm in a good mood.
   
   
   
6. All trees have leaves.
   Roses have leaves.
   Therefore, roses are trees.