Consider the following languages : $$$\eqalign{ & {L_1} = \left\{ {{a^n}{b^m}{c^{n + m}}:m,n \ge 1} \right\} \cr & {L_2} = \left\{ {{a^n}{b^n}{c^{2n}}:n \ge 1} \right\} \cr} $$$

Which one of the following is TRUE?

Consider the transition diagram of a $$PDA$$ given below with input alphabet $$\sum {\, = \left\{ {a,b} \right\}} $$ and stack alphabet $$\Gamma = \left\{ {X,Z} \right\}.$$ $$Z$$ is the initial stack symbol. Let $$L$$ denote the language accepted by the $$PDA.$$

Which one of the following is TRUE?

Consider the NPDA $$\left\langle {Q = \left\{ {{q_0},{q_1},{q_2}} \right\}} \right.,$$ $$\Sigma = \left \{ 0, 1 \right \},$$ $$\Gamma = \left \{ 0, 1, \perp \right \},$$ $$\delta, q_{0}, \perp,$$ $$\left. {F = \left\{ {{q_2}} \right\}} \right\rangle $$ , where (as per usual convention) $$Q$$ is the set of states, $$\Sigma$$ is the input alphabet, $$\Gamma$$ is the stack alphabet, $$\delta $$ is the state transition function q₀ is the initial state, $$\perp$$ is the initial stack symbol, and F is the set of accepting states. The state transition is as follows:

Which one of the following sequences must follow the string 101100 so that the overall string is accepted by the automaton?

Consider the DFAs M and N given above. The number of states in a minimal DFA that accepts the language L(M) ∩ L(N) is___________.