Discrete Mathematics Answers

a)Consider the decimal number a = 137.
i.Find in set builder notation the set of all positive integers b such that b ≡ a (mod 5).
ii.Is the number a prime? Explain.
iii.Convert the number a to binary and octal numbers

Let R be the partial order relation defined on A = {2, 3, 4, 5, 6, 8, 10, 40}, where xRy means x | y.
i.Draw the Hasse diagram for R.
ii.Find the upper and lower bounds of {4, 8}.

Find the smallest equivalence relation on {1, 2, 3} that contains (1, 2) and (2, 3).

Let G be a graph with 100 vertices numbered 1 to 100. Two vertices i and j are adjacent if |i-j|=8 or |i-j|=12. The number of connected components in G is
a)8
b)12
c)25
d)4

How many solutions are there to the equation
x1+x2+x3+x4+x5=21,
where xi, i=1,2,3,4,5, is a nonnegative integer such that
a)x1>=1?
b)xi>=2, for i=1,2,3,4,5?
c)0<=x1<=10?
d)0<=x1<=3, 1<=x2<4, and x3>=15?

In how many ways can a set of two positive integers less than 100 be chosen?

How many different strings can be made from the letters in ORONO, using some or all of the letters?

Find the solution to the recurrence relation a_n=〖-3a〗_(n-1) 〖-3a〗_(n-2) 〖-a〗_(n-3) with initial conditions a_0=1,a_1=-2 and a_2=-1

A binary relation on a set that is reflexive and symmetric is called a compatible relation. Let A be a set. A cover of A is a set of non-empty subsets of A, say { A_(1,) A_2,A_3……A_n} such that union of A_i's is equal to A. Suggest a way to define a compatible relation on A from a cover of A.

From an initial position on the left bank of a river, a boatman is to transport a wolf, a goat and a
cabbage to the right bank of the river. His boat is only big enough to hold himself and one other object at a time. The wolf cannot be left alone with the goat, and the goat cannot be left alone with the cabbage. How should the boatman proceed? Describe a graph that can be used to solve the puzzle, and find a solution.