13. Show that the language L= {a" :n > 0, n ± 4} is regular.

2. What is the TG corresponding to the language of strings, over alphabet {a, b}, that are composed exactly of odd repetition of the substring ab, i.e. sample strings of this language are ab, ababab, ababab, etc.

Suppose that relations R and S have n tuples and m tuples respectively. What is the minimum number of tuples that the results of the following expression can have? oc(R) × S, for some condition C. A. n*m B. min{n,m} С.n D. 0 E. m F. n-m G. n+m H. max{n,m}

Let language L over alphabet Σ = { a, b } be defined as L = { a'b* }, where i is a multiple of some given n and k is a multiple of some given m, all non- negative integers. In other words: L contains all strings consisting of: o A number of a's that is a multiple of n o followed by a number of b's that is a multiple of m. Lis Using a constructive method, show that for any given values of n and m,. regular.

Consider the language L of strings over the alphabet Σ = {a, b} which contain an equal number of a’s and b’s, and the set of b’s is consecutive in the string, e.g., ab, abba, aabb, abbbaa, aaabbbbbaa, etc. Note that ε ∈ L for completeness (0 number of a’s is equal to 0 number of b’s, which are also consecutive). Show that this language is context-free by giving a context-free grammar. Please make sure to test the answers onhttps://web.stanford.edu/class/archive/cs/cs103/cs103.1156/tools/cfg/

Let language A= {w | w can be written ajbk where k is a multiple of j.}. a. Show that Ais not regular.

Prove that the following languages are not regular using the Pumping lemma and/or closure properties. Note: Do not use other methods. a. L1 = {w ∈ {a, b}∗: na(w) + nb(w) = 3 · nb(w)} (na (w) is used to denote the number of a's in the string w) b. L2 = {anbmck: n, m, k ∈ ℕ, n < m or n < k}