Let G = ({S, C}. (a, b), P, S), where P consists of SaCa, CaCab. Find L(G).
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- ) Let O be the set of odd numbers and O’ = {1, 5, 9, 13, 17, ...} be its subset. Definethe bijections, f and g as:f : O 6 O’, f(d) = 2d - 1, d 0 O.g : ø 6 O, g(n) = 2n + 1, n 0 ø.Using only the concept of function composition, can there be a bijective map from øto O’? If so, compute it. If not, explain in details why not...................................................................................................................................... [2+8]b) A Sesotho word cannot begin with of the following letters of alphabet: D, G, V, W,X, Y and Z.We define the relation: A Sesotho word x is related to another Sesotho word y if xbegins with the same letter as y.Determine whether or not this is an equivalence relation.If it is an equivalence relation then1. Compute C(sekatana)2. How many equivalence classes are there in all, and why?3. What is the partition of the English words under this relation?If it is NOT an equivalence relation then explain in details why it is notGiven g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.Let S be a set and let C = (π1, π2,...,πn) be an increasing chain ofpartitions (PART(S), ≤) such that π1 = αS and πn = ωS. Then, the collection HC = ni=1 πi that consists of the blocks of all partitions in the chain is a hierarchy on S.
- Order the following functions by asymptotics with respect to Ω. That is, find an orderingf1 , f2 , · · ·, of the following functions such that f1 = Ω(f2), f2 = Ω(f3) and so on.n2, (√2)logn , n!, log(n)!, (3/2)n , n3, log2n, loglogn, 4logn, 2n, nlogn, 2logn , 2√2logn , log(n!)Consider P, the set of palindromes over Σ, defined recursively as follows: Basis: λ ∈ P and ∀c ∈ Σ (c ∈ P) Recursive: ∀w ∈ P, c ∈ Σ (cwc ∈ P) Use structural induction to prove that r(w) = w for all w ∈ P.Given a function f : A → B, we get a corresponding function f : P(A) → P(B), that maps a subset X ⊆ A to its image f(X) ⊆ B. What condition must f : A → B satisfy for f : P(A) →P(B) to be one-to-one? What condition must f : A → B satisfy for f : P(A) → P(B) to be onto?
- Let the Universal set be {a,b,c,d,e,f,g,h,i,j] Consider the subsets A = {a,d,h,i} B= {b,c,d.,f,i,j} C = {a,c,e,f,g,} What is (An B' )'?Let M, represent the matrix representation of a certain relation R on the set A. Consider a set A with n distinct elements. M, would therefore be a n x n matrix. How many 1s and 0s will M contain if R is a rooted (directed) tree?Let A be the set {1,3,5,7,9} and B be the set {1,2,4,8} . Find |A⋃B|