L1 is a recursively enumerable language over Σ. An algorithm A effectively enumerates its words as w1, w2, w3, ... Define another language L2 over Σ Union {#} as {wi # wj : wi, wj ∈ L1, i < j}. Here # is a new symbol. Consider the following assertions.
S1 : L1 is recursive implies L2 is recursive
S2 : L2 is recursive implies L1 is recursive
Which of the following statements is true ?
Question 9-Explanation:
S1 is TRUE.
If L1 is recursive L2 must also be recursive. Because to check if a word w=wi#wj belongs to L2, we can give wi and wj to the decider for L1 and if both are accepted then w belongs to L1 and not otherwise.
S2 is TRUE.
With a decider for L2 we can make a decider for L1 as follows. Let w1 be the first string enumerated by algorithm A for L1. Now, to check if a word w belongs to L1, make a string w′=w1#w and give it to the decider for L2 and if accepted, then w belongs to L1 and not otherwise.
So, the answer must be (A).