1. ## Direct Proof Question

The problem: Let x be an element of the integers. If 2^(2x) is an odd integer, then 4^x is an odd integer.

This problem was different from the previous direct proof problems we were assigned and I'm not sure if the phraseology is correct or not. Here is my proof:

"Let x be an element of the integers. Assume 2^(2x) is an odd integer. Obviously 2^(2x) = 4^x. For x < 0, 4^x is neither even nor odd. For x greater than or equal to 0, 4^x is even except when x=0. Since 0 is an element of the integers it follows that 4^x is odd."

2. Originally Posted by nick898
The problem: Let x be an element of the integers. If 2^(2x) is an odd integer, then 4^x is an odd integer.

This problem was different from the previous direct proof problems we were assigned and I'm not sure if the phraseology is correct or not. Here is my proof:

"Let x be an element of the integers. Assume 2^(2x) is an odd integer. Obviously 2^(2x) = 4^x. For x < 0, 4^x is neither even nor odd. For x greater than or equal to 0, 4^x is even except when x=0. Since 0 is an element of the integers it follows that 4^x is odd."
DO you want to prove that 4^x is even for all xεN instead of odd??

3. Nick, your proof is just fine for the problem as stated (regardless how dumb I think it is).
If $\displaystyle 2^x \text{ is odd}\; \Rightarrow \;x = 0$.

4. I understand what you're saying, but I asked my professor about it and he said that using x=0 was correct. I just needed to make the assertions that I had outlined. (My assertion being x=0 would come into play.) I just was confused over the phrases I'm using. Specifically this part:

"For x < 0, 4^x is neither even nor odd. For x greater than or equal to 0, 4^x is even except when x=0."

Would that be correct to assert that?