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.

Answers:

1. All nerds are good at math.
   Buffy is good at math
     Buffy is a nerd.
   
   Invalid by converse error.
   
   
   
2. It is difficult to study whenever I am tired.
   I found it easy to study today.
      I am not tired today.
   
   Valid by universal modus tollens.
   
   
   
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.
   
   Valid by universal modus tollens.
   
   
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.
   
   Invalid by inverse error.
   
   
   
5. I'm always in a good mood when the weather is good.
   The weather today is great.
     I'm in a good mood.
   
   Valid by universal modus ponens
   
   
   
6. All trees have leaves.
   Roses have leaves.
   Therefore, roses are trees.
   
   Invalid by converse error.