Question

Define a set T rocursively as follows: 1. BASE: λ∈T II. RECURSION: If x∈T, then 1×10∈T and ×0∈T III. RESTRICTION: Nothing is in Tother than the objects defined in I and II above. Find all strings in T with a length ≤5.

Answer 1

A set T is defined recursively to include the empty string λ as a base case, and then expanded by appending '10' or '0' to existing strings in T. All strings within the set T of length ≤5 are constructed iteratively using the given rules and include strings like λ, 0, 10, 00, 100, 010, and variations thereof up to length 5.

Step-by-step explanation:

The student is asking to define a set T recursively to find all strings within it that are of length less than or equal to 5. The set is defined with a base case, recursive step, and a restriction. Starting with the base case, we have the empty string λ as an element of T. From the recursive step, if x is an element of T, we can generate new elements by appending '10' to x (written as 1×10), or by appending '0' to x (written as ×0). Considering the restriction, we're only interested in elements that can be constructed from these rules.

Below is the step-by-step constructed set of T for strings of length ≤5:

1. λ (length 0)
2. 0 (length 1)
3. 10 (length 2)
4. 00 (length 2)
5. 100 (length 3)
6. 010 (length 3)
7. 000 (length 3)
8. 1000 (length 4)
9. 0100 (length 4)
10. 0010 (length 4)
11. 0000 (length 4)
12. 10000 (length 5)
13. 01000 (length 5)
14. 00100 (length 5)
15. 00010 (length 5)
16. 00000 (length 5)