Algorithm for Membership Testing A group G working on [2 = 1, 2..... n, a permutation g of 2 = 1, 2..... n, a base and strong generating set forG, sets U (i),1 I k of coset representatives for stabiliser chain, are the inputs. a boolean value that indicates whether or not g E G;

Exercise c. Strict matchingConsider the following generalization of the maximum matching problem, which we callStrict-Matching. Recall that a matching in an undirected graph G = (V, E) is a setof edges such that no distinct pair of edges {a, b} and {c, d} have endpoints that areequal: {a, b} ∩ {c, d} = ∅. Say that a strict matching is matching with the propertythat no pair of distinct edges have endpoints that are connected by an edge: {a, c} ̸∈ E,{a, d} ̸∈ E, {b, c} ̸∈ E, and {b, d} ̸∈ E. (Since a strict matching is also a matching, wealso require {a, b} ∩ {c, d} = ∅.) The problem Strict-Matching is then given a graphG and an integer k, does G contain a strict matching with at least k edges.Prove that Strict-Matching is NP-complete.

4. Problem Let T be a set and partially order P(T) by inclusion. Let C ≤ P(P(T)) be a chain and let F be a finite subset of Uuec U Prove that there is some U E C such that F C U. Hint: Clearly this is an induction problem.

On S = f2; 4; 6; 10; 12; 20; 30; 60g we work with the divisibility relation R:xRy , x divides y:(i) Draw the Hasse diagram of R.(ii) Find all R-minimal elements.(iii) Find all R-maximal elements.(iv) Find all non-empty subsets U  S, so that 6 is an R-lowerbound of U.(v) Let A = f6; 10; 12g. If A admits an R-supremum,  nd it. If A admits an R-in mum,  nd it.(vi) Is R a lattice?

9. Limitation Question. Using the idea of Pumping Lemma, please explain, why we can't build a Finite Automaton to generate all and only strings of the form 1³n0n where n>=0.