# Discrete Mathematics Problems Worth Solving, Questions and Answers

- 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.