Question

Let L be {w ∈ {0, 1}∗: w has an even number of zeroes }.

True or false:
(a) 00 ≈L 0
(b) 001 ≈L 010
(c) 10010 ≈L 101

Answer 1

In this case, option (a) is true, option (b) is false, and option (c) is true.

Step-by-step explanation:

In this case, the language L refers to the set of all binary strings that contain an even number of zeroes. Let's analyze each option:

(a) 00 ≈ L: This statement is true. The string 00 contains an even number of zeroes, so it belongs to L.

(b) 001 ≈ L: This statement is false. The string 001 contains an odd number of zeroes (1 zero), so it does not belong to L.

(c) 10010 ≈ L: This statement is true. The string 10010 contains an even number of zeroes, so it belongs to L.

For the pair (a) '00' and '0', '00' has an even number of zeroes (two 0s) hence is in L, while '0' has an odd number of zeroes (one 0) so it's not in L, thus '00' is not equivalent to '0' concerning language L. In pair (b) '001' and '010', both strings have an even number of 0s (two 0s each), therefore, they are equivalent concerning language L. Lastly, for pair (c) '10010' and '101', '10010' has an even number of 0s, thus is in L, while ‘101’ has one 0 which is odd, so ‘10010’ is not equivalent to ‘101’ in terms of language L.