High School

Let \( M_1 \) and \( M_2 \) be the finite automata in the figure below for the languages \( L_1 \) and \( L_2 \).

Find \( L_1 \cup L_2 \) and \( L_1 \cap L_2 \).

Option 1: \( L_1 \cup L_2 = \{0, 1, 00, 11, 000, 111, 001, 010, 011, 100, 101, 110\} \)

Option 2: \( L_1 \cap L_2 = \{0, 1, 00, 11, 000, 111, 001, 010, 011, 100, 101, 110\} \)

Option 3: \( L_1 \cup L_2 = \{0, 1, 00, 11\} \)

Option 4: \( L_1 \cap L_2 = \{0, 1\} \)

Answer :

Final answer:

The union of L1 and L2 is Option 1: {0, 1, 00, 11, 000, 111, 001, 010, 011, 100, 101, 110}. The intersection of L1 and L2 is Option 4: {0, 1}.

Explanation:

To find the union (L1 U L2) of two languages, we need to include all the words that are in either L1 or L2, or both. From the options provided, Option 1: L1 U L2 = {0, 1, 00, 11, 000, 111, 001, 010, 011, 100, 101, 110} is the correct answer as it includes all the words that are in L1 and L2. To find the intersection (L1 ∩ L2), we need to include only the words that are common to both L1 and L2. Option 4: L1 ∩ L2 = {0, 1} is the correct answer as it includes only the words 0 and 1 that are common to both L1 and L2.

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