1. © M. Winter COSC/MATH 4P61 - Theory of Computation 3.13.1 Pumping Lemma as a game 1.Player 1 picks the language L to be proven nonregular. 2.Player 2 picks n secretly. 3.Player 1 picks w (may depend on n). 4.Player 2 divides w into x, y, and z so that y≠ε and |xy|≤n. 5.Player 1 wins by picking k (may depend on n, |x|, |y|, and |z|) so that xy k z is not in L.

2. © M. Winter COSC/MATH 4P61 - Theory of Computation 3.23.2 1.The union of two regular languages is regular. 2.The intersection of two regular languages is regular. 3.The complement of a regular language is regular. 4.The difference of two regular languages is regular. 5.The reversal of a regular language is regular. 6.The closure (by *) of a regular language is regular. 7.The concatenation of two regular languages is regular. 8.A homomorphic image of a regular language is regular. 9.An inverse homomorphic image of a regular language is regular. Closure Properties of Regular Languages

3. © M. Winter COSC/MATH 4P61 - Theory of Computation 3.33.3 Example I

4. © M. Winter COSC/MATH 4P61 - Theory of Computation 3.43.4 Example II

5. © M. Winter COSC/MATH 4P61 - Theory of Computation 3.53.5 Example I (continued)