# Excellent Solutions to Computer Science Topics and Concepts

Get excellent solutions to different topics and subtopics in computer science. Explore solutions to questions on algorithms, data structures, discrete mathematics, operating systems, and a lot more.

- Problem of recursion (Discrete Math) Assume that changing the temperature of an object during a time interval is proportional to the difference in temperature between the object and the environment. A piece of metal, originally at 100…
- Which of the following statement are True.(1) x>0 is necessary for x+2>1?
- Suppose we model the spread of a virus in a certain population as follows. On day 1, one person is infected. On each subsequent day, each infected person gives the cold to two others. (a) Write down a recurrence relation for this mod…
- Suppose we model the spread of a virus in a certain population as follows. On day 1, one person is infected. On each subsequent day, each infected person gives the cold to two others. (a) Write down a recurrence relation for this mod…
- Draw the venn diagram of sets A, B and C where A is contained in (subset equal to) B , A intersection B doesn't equal to phi (empty set) , B intersection C = phi ( empty set) . What is the universal set you have chosen? Justify your c…
- Let X be a finite set with |X| > 1. What is the difference between P1 = X ×X and P2 = {S ∈ P(X) | |S| = 2}? Which set, P1 or P2, has more elements?
- A committee of three is chosen from a group of 20 people. How many different committees are possible, if (a) the committee consists of a president, vice president, and treasurer? (b) there is no distinction among the three members o…
- f is the function from {a, b, c} to {1, 2, 3} such that f(a)=2, f(b)=3, f(c)=1. Is f invertible, and if it is, what is its inverse?
- Let f be the function from {a, b, c, d} to {1, 2, 3} defined by f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto function?
- Find the number of circular 3-permutations of 5 people.
- Prove that A ∩ B = A ∪ B.
- Question five For each of the following sentences, write the sentence in logical notation , negate the sentence , and say whether the sentence or its negation is true. a) Given any integer, there is a larger integer.
- Question one Prove by induction that the following statements are true for all integers n a) 12x2+22x3+…..+n2(n+1)= n(n+1)(n+2)(3n+1)/12 b)4007n-1 is divisible by 2003
- X be a non-empty set and let R be an equivalence relation on X. For each x ∈ X, define [x]={y∈X suchthatxRy} to be the equivalence class of x. Here x R y means (x, y) ∈ R. SupposethatA=[x]andB=[y]. ProvethatifA∩B̸=∅,thenA=B.
- Find domain and range of (answers should be subsets of R): f(x)= 3/2x -1
- Find domain and range of (answers should be subsets of R): f(x)= 3/2x -1
- Find domain and range of (answers should be subsets of R): f(x)= 1/(5x-6)
- Find domain and range of (answers should be subsets of R): f(x)= x/(x^2+1)
- Find domain and range of (answers should be subsets of R): f(x)= 3/(2x -1)
- 3. Prove or give a counterexample to the following: For a set A and binary relation R on A, if R is reflexive and symmetric, then R must be transitive as well.
- Let X = {1,2,3}. Define a relation ∼ on P(X) by A ∼ B if A and B have the same number of elements. Prove that ∼ is an equivalence relation. Write down all equivalence classes of ∼.
- Let X be a non-empty set, and let R be an equivalence relation on X. Let C be the set of all equivalence classes of R. So C={A⊆X such that A=[x] for some x ∈ X}. Now, define f : X → C by the rule f(x) = [x] for all x ∈ X. Prove that…
- Let X be a non-empty set, and let R be an equivalence relation on X. Let C be the set of all equivalence classes of R. So C={A⊆X such that A=[x] for some x ∈ X}. Now, define f : X → C by the rule f(x) = [x] for all x ∈ X. Suppose X …
- Let X = {1,2,3}. Define a relation ∼ on P(X) by A ∼ B if A and B have the same number of elements. Prove that ∼ is an equivalence relation and write down all equivalence classes of ∼.
- How many ways can you assign 4 caretakers each of which look after 2 lighthouses for a year (for a total of 8 lighthouses), and then assign each two lighthouses in the next year so that none of them tend the same two lighthouses both …
- Find a generating function in closed form for the sequence: {1,2,3,4,1,2,3,4,1,2,3,4,...}
- Prove that for all sets A and B: A⊆B ⇐⇒ A∪B=B.
- Prove that A − (B ∪ C) = (A − B) ∩ (A − C), for sets A, B and C.
- Prove that for all sets A and B, A ∪ B^c = B^c ∪ (A ∩ B).
- Prove that A − (B ∪ C) = (A − B) ∩ (A − C),
- Prove that 3n2 − n + 10 is an even integer for all integers n. (Hint: Prove the statement true when n is odd, then prove it is true when n is even.)
- Define the relation R as R = {(a, b)|a, b ∈ Z, 4 divides a − b}. Show that R is reflexive, transitive and symmetric.
- Let X = {0, 1, 2}. Consider the power set P(X). Recall that P(X) = {A such that A ⊆ X}. Determine if each statement is true or false. (a) X ∈ P(X). (b) ∅∈P(X) (c) X ⊆ P(X). (d) {0,2}∈P(X) (e) {0,2,1} ∈ X (f) {0}⊆X (g) {{0}} ⊆ P(X)
- Prove that for all positive integers n 3^n +7^n −2 is divisible by 8
- Prove that for all integers n with 1≤n≤10,n^2 −n+11 is prime.
- Describe two situations that can be modelled with a Graph. Describe the graph model that you will use
- Describe two situations that can be modelled with a Tree. Describe the graph model that you will use.
- Describe two situations that can be modelled with a Graph. Describe the graph model that you will use. Describe two situations that can be modelled with a Tree. Describe the tree model that you will use.
- Let R = {(2, 2), (2, 4), (3, 3), (3, 6), (3, 12), (4,2), (6, 3)} where if a is a factor b} (i) Draw the digraph for the relation P. (ii) Find the domain and range of the relation.
- Q:1. Determine whether each set is a function from X = {1,2,3,4} to Y = {a,b,c,d}. If it is a function, find its domain and range, draw its arrow diagram, and determine if it is one-to-one, onto or both. a) {(1,a,),(2,a),(3,c),(4,b)} …
- Q : 1) Determine whether each function is one-to-one. The domain of each function is the set of all real numbers. If the function is not one-to-one, prove it. Also, determine whether f is onto the set of all real numbers. If f is not …
- Q : 5). If f is function from A to B and g is function B to C and both f and g are onto. Show that go f is also onto. Is go f one-to-one if both f and g are one-to-one. Q : 6) Let f, g and h: R → R be defined by (R is the set of rea…
- Construct the call graph for a set of seven telephone numbers 555-0011, 555-1221, 555-1333, 555-8888, 555-2222, 555-0091, and 555-1200 if there were three calls from 555-0011 to 555-8888 and two calls from 555-8888 to 555-0011, two ca…
- Discrete Mathematics - Languages and Machines Need to solve Finite State Automaton i.e, Your task is to design a binary finite state automaton (FSA) to accept all strings that represent valid messages (for your particular codes and pa…
- For any two sets A and B , in a universal set U , prove that A ⊆ B ⇔ A ∪ B = B.
- a) Simplify the Boolean function F = AB + (AC)′ + AB ′C(AB + C).
- Implement the 3-variable function F (A,B,C) = (0,2,4,7) with a multiplexer
- a. Minimize the following problems using the Karnaugh maps method. Z = f (A, B, C) = + B + AB + AC
- Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).
- (A \ B )∪ C = A \ (B∩ C )for any three sets A,B and C Is the statement true or false? Give justification in support of your answer.