**What is recursively enumerable language?**

If any Turing Machine can be designed to accept all string of the given language, then the language is called recursively enumerable language. RE languages are the formal languages.

[quads id=1]Properties of Recursively enumerable languages:

- Union
- Intersection
- Complement

**1.Union of RE languages:**

Lets revise union of sets;

Set 1={a, b, c}

Set 2={b, c, d}

Set 1 Union Set 2 = {a, b, c, d}

Now let’s understand the same concept in Turing Machine;

[quads id=2]- Suppose a system has 2 Turing Machines, TM1, and TM2.
- If TM1 halts then all the system halts.
- If TM1 crash then system checks that TM2 is ready to halt or not? If TM2 halts then system halts because this is union and the union means that
- If TM1 halts then system halts
- If TM1 does not halt, and TM2 halts then system halts
- If TM1 and TM2 or TM
_{n}halts then system halts

**2. The intersection of RE languages:**

Let’s revise intersection of sets;

Set 1={a, b, c}

Set 2={b, c, d}

Set 1 Intersection Set 2 = {b, c}

Now let’s understand the same concept in Turing Machine;

- Suppose a system has 2 Turing Machines, TM1, and TM2.
- If TM1 crash then all the system crash.
- If TM1 halts then system checks that TM2 is ready to halt or not? After this, If TM2 halts then system halts because this is intersection and the intersection means that
- If TM1 crash then system crash
- If TM1 halts then check TM2 or TM
_{n}, and if TM2 is also halted, then system halts. - If TM1 and TM2 or TM
_{n}crash then the system crash

**2. The complement of RE languages:**

[quads id=3]- Suppose a system has 2 Turing Machines, TM1, and TM2.
- If TM1 crash then all the system crash.
- If TM1 halts then system check TM2 or TM
_{n}. If TM1 halts and TM2 also halts then system crash. - If TM1 halts then system check TM2 or TM
_{n}. If TM1 halts and TM2 crash then system halts.