Let S: e e, S(a1, a2, .) = (0, a1, a2, . ) %3D be the forward shift operator. Show that the adjoint operator S* is the backward shift S*: ( e, S*(a1, a2, . .) = (a2, a3, - . .). Verify that S*S = I, but SS + I.

Let ü= Compute ū - v. -5,6

In (Z24, €) find the orders of the elements 3, 4, 6, 7, and 18.

Verify the relations (5).

What is the similarity transformation that maps the preimage (dark line triangle) to the image (dotted line triangle)?

Assume that U= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }, A = { 2, 5, 7, 8 } and B ={ 1, 4, 5, 7 }. Find (AU B) '. * O (A U B) '= { 3, 6, 9, 10} O (A U B)'= { 1, 2, 4, 5, 7, 8 } O (A U B)' = { 5, 7 } O (A U B)'= { 1, 2, 3, 4, 6, 8, 9, 10}

Given F(A, B, C, D) = A'B'C'D' + A'B'C'D + A'BCD' + A'BCD, express F(A,B,C,D) and F '(A,B,C,D) in minterm list form.

Let A={1,2,3,4,5} and R={(1,1),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5),(4,4),(5,5)} · Show that the relation R is a partial order · Draw Hasse diagram. · Identify Maximal, Minimal, Greatest, Latest elements. · identify Upper and Lower Bound along with Greatest Lower Bound, Least Upper Bound

Let f and g be permutations. Show that the order of h=fgf^−1 is the same as the order of g.

26 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.