Statistical hypothesis testing

[Tim Frink]

Hi,

I've a question about on statistical hypothesis testing. Let's say I've
two approaches A and B and I would like to see if one approach is better
than the other.

These are my results (p-values) for the statistical testing:

A B
A - 0.002
B 0.998 -

using the alternative hypothesis that the approaches in the first column
are better than the approaches in the 2. and 3. column. The significance
level was 5%.

Is my interpretation correct that statically significant differences are
observed in the second row, i.e. it can be inferred that A outperforms B
(since 0.002 < 5%)?

If on the other hand, the table would be a result of significance testing
using the null hypothesis that approaches in the first column perform as
well as approaches in the 2. and 3. column. Could it be then inferred
that
a) the null hypothesis can be rejected for 0.002 (since smaller than
significance level), i.e. A and B do not perform equally
b) the null hypothesis is valid for 0.998, i.e. B and A perform equally?

So, is the interpretation of the p-value w.r.t. to a given significance
level oppositional for the null and alternative hypothesis? Can it be
said in general that if the null hypothesis is rejected then the
alternative hypothesis can be accepted?

Regards,
Tim