Show that the set
*22. One important technique used to prove that certain sets not regular is the pumping lemma. The pumping lemma states that if
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DISCRETE MATHEMATICS LOOSELEAF W/CONNECT
- Exercises 7. Express each permutation in Exercise as a product of transpositions. 1. Express each permutation as a product of disjoint cycles and find the orbits of each permutation. a. b. c. d. e. f. g. h.arrow_forwardProve statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.arrow_forwardSuppose that A is an invertible matrix over and O is a zero matrix. Prove that if AX=O, then X=O.arrow_forward
- Find the order of each permutation in Exercise 1. Express each permutation as a product of disjoint cycles and find the orbits of each permutation. a. [ 1234545312 ] b. [ 1234513254 ] c. [ 1234541352 ] d. [ 1234535241 ] e. [ 12345673456127 ] f. [ 12345675137264 ] g. [ 1234513452 ][ 1234532415 ] h. [ 1234523415 ][ 1234513542 ]arrow_forwardConsider the following language over the alphabet Σ = {a, b} L = {w : nb(w) is even, na(w) is odd, and w does not contain the substring ba} 1. Design the minimal deterministic finite automata which recognizes L. You do not need to prove correctness nor minimality. 2. Provide regular expressions for each of the equivalence classes of ≡L. [Hint: Use your answer from 1. A conversion algorithm is not necessary.]arrow_forward11. (10 pts) Given a relation R from a set X = {1,..., m} to a set Y = (1,...,n}, we represent R as a matrix A by Aij 1 if (i, j) € R and Aj = 0 if i and j are not related. Write an algorithm to determine if relation R is a function from X to Y. =arrow_forward
- If Vo and V; are respectively the t-conorms that are dual to the t-norms Ao and Ag respectively. Show that for any t-conorm V.arrow_forwardGive all σ-algebra generated by S = {0, 2, 4, 5, 7}.arrow_forward26 of 40 Consider the Datalog programs P1 (left) and P2 (right) below, which use relations R(A, B) and S(A, B). P1 P2: T1(A) R(A, B). T4(A) R(A, B), S(A, B). T2(A) S(A, B). T3(A) + T1(A), T2(A). Which of the following statements is TRUE about the relationships between relations T3 and T4 defined by P1 and P2, respectively? Note that the commas "," used in the rule bodies to separate the predicates is the same as using AND. Select one: T3 and T4 include the same set of tuples. Every tuple in T3 is also contained in T4, that is, T3 C T4. O None of the other answers, that is, T3 and T4 contain different tuples, in general. O Every tuple in T4 is also contained in T3, that is, T4 C T3.arrow_forward
- Design a deterministic finite automaton that accepts all binary strings that correspond to a value divisible by 3. For example, it should accept 110 (since 6 is divisible by 3), but not 101 (since 5 is not divisible by 3).arrow_forwardLet set A = {1,2,3,4} and let R1 and R2 be binary relations on A. Specifically, let: R1 = {(1,1), (1,2), (2, 1), (2, 2), (2, 4), (3, 4), (4, 2), (4, 3) (4, 4)} R2 = {(1,2), (1, 3), (1, 4), (2, 1), (2,3), (4, 1), (4, 2)} Determine the following: a) Whether R, is reflexive, irreflexive, symmetric, anti-symmetric and/or transitive. b) Whether R, is reflexive, irreflexive, symmetric, anti-symmetric and/or transitive. c) R1 • R2. d) R2 • R1. e) R1 U R2. f) Rz n R2. g) The reflexive, symmetric, and transitive closures of both R, and R2arrow_forward{₁,...,} is linearly independent if and only if {₁,..., V} is an orthogonal set. True Falsearrow_forward
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