The correct statements are: If L₁ and L₂ are not regular languages, then L₁ U L₂ cannot be regular. If L₁ and L₂ are regular languages, then L₁ L₂ = {w: w€ (L₁-L₂) or w€ (L₂ - L₁)} is regular. It is possible that the union of a regular language and a nonregular language is regular. Every subset of a regular language is regular.

Let E be the alphabet {a,b}. E* – (e|E*b)aa(ɛ|b£*), where the hyphen denotes set difference and ɛ de- notes the empty string. (Hint: L consists of the strings over E in which no maximal substring consisting solely of a's is of length 2.) Draw a finite automa- ton that accepts L. Let L be the language described by

1. Is the following language B regular or non-regular? B = {0% 0 | a,b ≥ 0 and a =b} 2. If your answer above is regular, please provide a Regular Expression that recognizes the language. If your answer is non-regular, please prove it by contradiction via Pumping Lemma for Regular Languages. The format and style of the proof using Pumping Lemma are supposed to follow those in the textbook.

2. The function f is defined for non-negative integers a and b recursively as follows: f (а, b) 3D1 if a = 0 or b = 0 f (a - 1, b f (a f (а, а) 1) + 2a - 1 if a if a > b if a < b b b, b) + f (b, b) + f (b а, а) } Compute f (2, 3) by drawing a recursion tree showing all of the computation required and then use your tree to compute the answer.

Let C(x. v) be x is enrolled in y, where the domain for x is the set of all students at a certain university and the domain for y is the set of all computer science classes being offered at the university. Match each English statement with its equivalent logical expression. ? v 1. No one is enrolled in both CS222 and CS252. A. -3x(C(x, CS222) ^ C(x, CS252)) B. C(Sarah, CS232) C. Ex(C(x, CS222) v C(x, CS252)) D. Vx3yC(x, y). E. 3yVxC(x, y) F. -Vx(C(x, CS222) A C(x, CS252)) G. VyC(Sarah, y) H. 3XVYC(x, y) 1. 3x(C(x, CS222) A C(x, CS252)) J. ByC(Sarah, y) v 2. Everyone is enrolled in some computer science classes, ? v 3. Sarah is enrolled in some computer science class. ? v 4. Someone is enrolled in all domputer science classes. v 5. Someone is enrolled in both CS222 and CS252. ? v 6. There is one class that is enrolled by all students.

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/

Suppose LS {a, b}* is defined as follows: A e L; for every x and y in L, the strings axb, bxa, and xy are in L. Show that L= AE B, the language of all strings x in {a, b}* satisfying na(x) = n¿(x).

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