Answer
Let's solve the problem step by step.
a) Show that 111000 belongs to the language generated by G:
1. Start with the start symbol S.
2. Apply production 1: S → 1S → 11S
3. Apply production 1 again: 11S → 111S
4. Apply production 3: 111S → 11100A
5. Apply production 5: 11100A → 111000
Thus, 111000 can be derived from S using the productions in P, so it belongs to the language generated by G.
b) Show that 11001 does not belong to the language generated by G:
Assume, if possible, that 11001 belongs to the language generated by G.
1. Start with the start symbol S.
2. Apply production 1: S → 1S → 11S (to get the first two 1s)
3. Since there's no production to add a 0 between 1s, we cannot derive 11001.
Thus, 11001 cannot be derived from S using the productions in P, so it does not belong to the language generated by G.
c) Language generated by G:
From the productions, we can see that G generates strings of two types:
1. Strings starting with 1, followed by any number of 1s (including 0)
2. Strings starting with 00, followed by any number of 0s (including 0)
Let's represent this language mathematically:
L(G) = {1^m | m ≥ 1} ∪ {00^n | n ≥ 0}
Where 1^m represents a string of m 1s, and 00^n represents a string of n 0s preceded by 00.
In regular expression form, L(G) can be written as: (1+ | 00*)
Work Step by Step
a) 111000 belongs to L(G)
1. Start with S.
2. Use rule 1: S → 1S (add 1)
3. Repeat step 2: 1S → 11S → 111S
4. Use rule 3: 111S → 11100A
5. Use rule 5: 11100A → 111000
We made 111000 from S using the rules!
b) 11001 does NOT belong to L(G)
1. Try to make 11001 from S.
2. Use rule 1: S → 1S (add 1)
3. Repeat step 2: 1S → 11S
4. Problem! No rule adds 0 between 1s.
We can't make 11001 from S using the rules.
c) Language L(G)
L(G) has two types of strings:
1. Strings starting with 1 (then any number of 1s)
2. Strings starting with 00 (then any number of 0s)
Math formula: $L(G) = {1^m | m ≥ 1} ∪ {00^n | n ≥ 0}$
Regular expression: $(1+ | 00*)$
Think of it like a recipe:
• Start with 1 (and add more 1s)
• Or start with 00 (and add more 0s)
That's L(G)!
