Determine the following languages of Σ∗

Question

For the alphabet Σ = {0, 1}, let A, B, C ⊆ Σ∗ be the languages below.

i. A={1,0,00,11,000,111,0000,1111}

ii. B={w∈Σ∗|||w||≥2}

iii. C = {w ∈ Σ∗|||w|| ≤ 2}

Note: ||w|| denotes the length of the word w.

Determine the following languages of Σ∗:

a. A∩B.

b. A∩C.

c. B∪C.

d. A∆B.

Now what I have so far is,

a) A∩B = {00, 11, 000, 111, 0000, 1111}

b) A∩C = {1, 0, 00, 11}

c) B∪C. = Σ∗

d) A∆B. = the elements that are in A or B but NOT both.

Now first, I wasn't sure if I understood this properly... Second, I wasn't sure if the period symbol actually means anything relevant or if it is just there to act as a period. Third, I wasn't sure how to actually write out the answer for d because there are so many possible values that fit.

Answer 1

Personally I think the most elegant way to write (d) is $\Sigma \cup B$. But you can probably write it in any way that conveys that you understand what you're doing and uses correct notation.

And no, the period doesn't have mathematical meaning. It's just a period.

All the answers are correct.