High School

Prove that for two languages [tex]L_1[/tex] and [tex]L_2[/tex], if [tex]L_1 \subseteq L_2[/tex], then [tex]L_1^R \subseteq L_2^R[/tex].

Answer :

Final answer:

If a language L1 is a subset of another language L2, that means if we reverse every string in L1, every reversed string must also be in the reversed L2, proving L1R⊆L2R.

Explanation:

Let's start by defining what the symbols here represent in the context of formal language theory. The symbol '⊆' represents 'is a subset of', meaning every element in the first set is also an element in the second set. The symbol '⇒' represents 'implies', meaning if the first proposition is true, the second must also be true. 'R' represents the reverse of a language, meaning all words in the language are reversed.

If we say L1​⊆L2, we mean that every string that is in language L1 is also in language L2. Therefore, if we reverse every string in L1 (creating L1R), every reversed string must also be in the reversed version of L2 (L2R), because the original string before it was reversed was in L2. This gives us L1R​⊆L2R

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