L = { a^n b^n c^m d^m : n >= 1, m >= 1 } U { a^n b^m c^m d^n : n >= 1, m >= 1 } isRegular?

Question

there is a lot of examples for pumpinglemma proof, but I did not figure out this, can anyone help ?

L= { a^nb^nc^md^m : n >= 1, m >= 1 } U { a^nb^mc^md^n : n >= 1, m >= 1 }

Answer 1

Consider the regular language R = a*b*cd . The intersection of two regular languages must be a regular language. The intersection of L and R is a^nb^n cd . However, this is easily shown not to be regular using the pumping lemma or Myhill-Nerode theorem. This is a contradiction, so L must not be regular.