1. Using closure properties
(a) Which one of the following eight statements implies that the language LIS recursive?
[2, 2.0]
The correct answer is proof (v). Since a finite language is also solvable, we have that
both L1and L2are solvable. Since solvable langauges are closed under intersection, this
implies Lis solvable.
(b) Which one of the following eight statements implies that the language LIS NOT
recursive? Explain your reasoning. [2, 3.0]
The correct answer is proof (ii). Since L1is solvable, it follows that if Lis solvable, then
L2must also be solvable since solvable languages are closed under intersection. However,
we are given that L2is not solvable. Therefore, Lmust not be solvable.
i. We show that L1∩L2=Lwhere L1is a recursively enumerable but not recursive
language and L2is a recursively enumerable language.
ii. We show L1∩L=L2where L1is a recursive language and L2is not a recursively
enumerable language.
iii. We show that L1∩L2=Lwhere L1is a recursively enumerable language and L2is
not a recursive language.
iv. We show that L1∩L=L2where L1is a recursively enumerable language and L2is
not a recursive language.
v. We show that L1∩L2=Lwhere L1is a recursive language and L2is a finite
language.
vi. We show that L1∩L=L2where L1is a recursive language and L2is a recursive
language.
vii. We show that L1∩L2=Lwhere L1is a recursively enumerable language and L2is
a recursively enumerable language.
viii. We show that L1∩L=L2where L1is a recursive language and L2is a recursively
enumerable language.
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