Show that the language {a^n b^m a^n|n greater than or equal to 0} is not regular using the pumping lemma, also that the language {a^n!|n greater than or equal to 0} is not regular by using the pumping lemma. And finally show that the language F = {a^i b^j c^k | i, j, k greater than or equal to 0 and if i = 1 then j = k} is not regular. Show, however, that it satisfies the statement of the pumping lemma, i.e. there is a p such that all three conditions for the pumping lemma are met. Explain why this does not contradict the pumping lemma.