Show that the collection of Turing-recognizable languages is closed under the following operations:

1. Union: If $ L_1 $ and $ L_2 $ are Turing-recognizable, then $ L_1 \cup L_2 $ is Turing-recognizable.

2. Intersection: If $ L_1 $ and $ L_2 $ are Turing-recognizable, then $ L_1 \cap L_2 $ is Turing-recognizable.

3. Concatenation: If $ L_1 $ and $ L_2 $ are Turing-recognizable, then $ L_1L_2 $ is Turing-recognizable.

4. Kleene Star: If $ L $ is Turing-recognizable, then $ L^* $ is Turing-recognizable.

5. Homomorphism: If $ L \subseteq \Sigma^* $ is Turing-recognizable, then $ h(L) \subseteq \Gamma^* $ is Turing-recognizable, where $ h : \Sigma^* \rightarrow \Gamma^* $ is a homomorphism.

See section 1.2 of the following notes for the definition of homomorphism.

The collection of Turing-recognizable languages is indeed closed under the Union, Intersection, Concatenation, Kleene star, and Homomorphism operations due to their unique properties when interacting with Turing machines.

The collection of Turing-recognizable languages are indeed closed under the operations mentioned in your question. Here's why:

1. Union: If L1 and L2 are Turing-recognizable, then their union is also Turing-recognizable.

This is due to the fact that we could design a Turing machine that operates by simulating the Turing machines for L1 and L2 in parallel, accepting if either accepts.

2. Intersection: If L1 and L2 are Turing-recognizable, then their intersection is also Turing-recognizable.

A Turing machine could be designed to recognize the intersection of L1 and L2 by simulating both Turing machines in parallel and accept only if both machines accept.

3. Concatenation: If L1 and L2 are Turing-recognizable, then their concatenation is also Turing-recognizable.

We could create a Turing machine that operates by non-deterministically splitting the input and verifying if the first piece is in L1 and the second piece is in L2.

4. Kleene star: If L is Turing-recognizable, then L* is Turing-recognizable.

A way around an example about closure of Turing recognizable languages under Kleene star, is the Turing machine accepts strings y1, y2, ..., yn for some n >= 0, such that each yi belongs to L.

5. Homomorphism: If a language L is Turing-recognizable, then its homomorphism is Turing-recognizable.

This is because if we can decide membership in L, we can decide membership in h(L) by running the Turing machine for L on the preimage of the string under h.

Learn more about the topic of Turing-recognizable languages here:

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