Induction on i proves (w) = (w) for every string w and all i > 0. Use these tips: Your theory (uv) = vRuR for E* u, v.

Prove Pierce's law: For any propositions P and Q, the proposition ((P -+ Q) -> P) -+ P is a tautology

Let E={0,1}. Give a regular expression r over Σ such that L(r) = { w in Σ* | (w contains at least 4 symbols, that is, the length of w is at least 4) and (the first symbol of w is different from the last symbol of w) and (interior(w) is vorpal according to the following two definitions)}. For any string y containing at least 4 symbols, let interior (y) be the substring of y obtained by removing the first two symbols and the last two symbols of y. A string x over Σ is called vorpal if x satisfies one or more of the following conditions: • x is made up of exactly k 0-symbols where k modulo 4 is 3 (and no 1-symbols) ● x is made up of at least two 1-symbols (and no 0-symbols) ● x contains 011

| Consider the elliptic curve group based on the equation y? = x' + ax + b mod p where a = 4, b = 1, and p = 7. This curve contains the point P = (0, 1). The order of this elliptic curve group is the prime number 5, and therefore we can be sure that P is a primitive element. Another element in this group is Q = (0,6). The index of Q with respect to P is the least positive integer d such that Q = dP. What is d, the index of Q?

Prove that L2 = {w = {0,1}+ | w starts and ends with the same symbol} is regular. string of length 1 (e.g., w = 0) starts and ends with the same symbol.

Prove ((A --> B) C) is a proposition.

Formally prove that on? + 10n + 10 = (n²) using the definition of 2(:).

We consider certain strings of length 6 over the alphabet S = {A,B,C}.  We require that the string contains exactly two As, and exactly two B's, and exactly two C's.  How many such strings are there? That is, how many strings of length 6 over the alphabet {A,B,C} contains exactly two A's, two B's and two C's?

Regular Expression for the set of all even length strings with at most two 0’s? Show me step-by-step solution.

Let Σ={0,1}. Give a regular expression r over Σ such that L(r) = { w in Σ* | (w contains at least 2 symbols, that is, the length of w is at least 2) and (the first symbol of w is different from the last symbol of w) and (centre (w) is mimsy according to the following two definitions)}. For any string y containing at least 2 symbols, let centre (y) be the substring of y obtained by removing the first symbol and the last symbol of y. A string x over Σ is called mimsy if x is made up of either • at least two 0-symbols (and no 1-symbols), or an even number of 1-symbols (and no 0-symbols).