9

1.A known relationship mode of R (U, F), U= (A, B, C, D, E, G); F={AB->C, D->EG,C->A, BE->C, BC->D, CG->BD, ACD->B, CE->AG}; find the minimum cover of F,then use the Normal Form decomposition algorithm to decompose this mode to 3NF, andcheck if the decomposition has the nonadditive join property.

1. A known relationship mode of R (U, F), U= (A, B, C, E, F, G, H, I, J, G); F={AB->E,ABE->FG, B->FI, C->J, CJ->I, G->H}; find the minimum cover of F, then use theNormal Form decomposition algorithm to decompose this mode to 3NF, and check if thedecomposition has the nonadditive join property.

Discrete Structures for CS

Distinguish among direct mapping, associative mapping, and set-associative mapping.

Discrete Mathematics  Simplify each of the following expressions. Do not use the set difference operator in your simplifications. Note that the set U in the second item is the universal set, which includes all elements. Shade the indicated regions of the following Venn diagrams.   please see attached photo for more details

Given the following UML diagram: Student - studentID: String + getStudentID(): String + toString(): String Person - lastName: String - firstName: String - dateOfBirth: Date - address: Address + getName(): String + getDOB(): Date + getAddress(): Address + toString(): String Professor - employeeID: String +getEmployelD(): String +toString(): String Write an implementation for the class Professor. DO NOT include or implement any constructor for this class. There are no abstract methods/classes in this example. toString method will return any simple String that contains firstName and employeelD only; no less, no more!! Assume firstName was defined as protected in class Person. You are not allowed to add implementation for any class or methods other than what is listed in the question statement.

b) Express the following relation using a set-builder notation R = {(1, 2), (2, 6), (3,9), (4, 16), (5, 20), (6, 36), ...}

What goes into the construction of an ADT? Your answer should fill up the gaps.

edit and add inter-class relations to the diagram

1. Consider the relation R on the set (a,b.c,d,e,f] which is defined as follows: R-{(a,a), (a,b). (a.c). (b.a). (b.c), (c,a). (c,b). (d.d), (f.f)}. Is R reflexive? Is R symmetric? Is R anti-symmetric? Is R transitive? Give reasons!

PLEASE TYPE ONLY**** ANSWER ALL THREE QUESTIONS Exercise 8.1.4: Arrow diagrams for relations on small finite sets.   Draw the arrow diagram for each relation. (b) The domain of relation P is {2, 4, 8, 16, 32, 64}. For x, y in the domain, xPy if there is a positive integer n such that xn = y. (c) The domain of relation D is {2, 3, 12, 16, 27, 48}. For x, y in the domain, xDy if y is an integer multiple of x. (d) The domain for the relation A is the set {2, 5, 7, 8, 11}. For x, y in the domain, xAy if |x-y| is less than 2

3. Draw ALL the possible parse trees for (p →-q → g) → (p → g), without assuming any binding priorities of the connectives. Then, for each of them, list all the sub formulas.

Given: Relation R = {(a,a), (a.b), (b,a). (b,b). (c,c)} and Set S = {a, b, c} Complete Relation R to be Transitive with Set S

Let A ={1, 2, 3, ... , 10}, B = {2, 3, 6, 8}, and C = {3, 5, 4, 8, 2} Write the Venn diagram representation of the following sets. How many elements are there in A U B U C.

i.First write a decomposition for the extended problem, based on the description in Box 1 above. This should be at a similar level of detail to the one given in the discussion to Activity 2.21 of Block 3 Part 2. The first sub-problem should be to set up a glossary, as before, but the rest of the decomposition needs modifying to reflect the extension to the problem. ii.Expand the part of the algorithm that deals with user interaction by instantiating the interactive loop Pattern 2.1 from Block 3 Part 2. You can see an example of this pattern being instantiated in the discussion for Activity 2.23; although, of course, your instantiation will be slightly different because the problem has changed. You can assume that, when asked if they know a definition, the user will always answer either y or n. iii.Briefly discuss how you will test the program. Only a short answer is required and you do not need to give examples, only describe what approach you will follow. iv.Now you will…

Let R be the relation on the set {a,b,c,d} containing the ordered pairs (a,a), (a,c), (b,b), (b,c), (b,d), (c,b), (c,c), (d,a) and (d,d). Find R².

Show that the following decomposition of the schema R of Exercise 7.1 is not alossless decomposition:(A, B, C)(C, D, E).Hint: Give an example of a relation r(R) such that ΠA, B, C (r) ⋈ ΠC, D, E (r) ≠ r

The given figure shows the module decomposition of a Use set to represent the grouped modules for integration testing, e.g. {(a, b) implies modules a and b are tested together. Which one is correct? 3 5 4 7 8 9 The bottom-up integration should start with (3.5). The bottom-up integration should start with [1,2,6). The bottom-up integration should start with (1,6). The bottom-up integration should start with (2,3,4,5). O The bottom-up integration should start with (6,7,8,9).

def transitiveclosure(g):     """computes the transitive closure of a graph/relation encoded as as a set of 2-tuples"""   transitiveclosure: This function should accept sets/frozensets of 2-tuples that encode relations, or equivalently, graphs, and should return the least relation that includes the input relation and is transitive: its transitive closure. E.g., an input frozenset({(1,2),(2,3)}) could correctly result in the value frozenset({(1,2),(2,3),(1,3)}).

Question 2 12 14 Consider the given node-based data structure. You will need to create a suitable abstract data type representation of the structure and process it according to the description given below. Each arm is a linked list attached to a central hub. Sub-question Description The data structure must be defined using only the basic types of your chosen programming language. No libraries may be used for the oreation of the type including those provided as standard in your programming language. Libraries may be used for input / output and all other purposes apart from defining the data structure. Q2-A Your program must define the structure. The number of arms should be flexible. Users should be able to insert values at the end of a given arm as well to display the contents of the structure as text. You do not need to support any other operations. Q2-B The nodes contain single integer values.

There is a great deal of difficulty in understanding abstract data types (ADTS). Does the use of ADTs offer any benefits? What role does the notion of heredity have in resolving problems?

Construct an NPDA that accepts L(G) for the following G:

Create a Venn diagram for each of the sets below. 1.  (A ∩ B ∩ C) 2. (A∪B) ∩ C

B)Distinguish among direct mapping, associative mapping, and set-associative mapping.