Let S = {1,2,3,...,10}.
a. Find the number of subsets of S that contain the number
5.
b. Find the number of subsets of S that contain neither 5 nor
6.
c. Find the number of subsets of S that contain both 5 and
6.
d. Find the number of subsets of S that contain no odd
numbers.
e. Find the number of subsets of S that contain exactly three
elements.
f. Find the number of subsets of S that...
Discrete Mathematics
A tree contains 1 vertex of degree 2, 1 vertex of degree 3, 1
vertex of degree 4, 11 leaves and the remaining vertices have
degree 3.
Find the total number of vertices.
Sketch two non-isomorphic trees statisfying the above
mentioned conditions.
MAT 204 Discrete Structures – Assignment #10
Number theory is the branch of mathematics concerned with the
integers. Traditionally, number theory was a pure branch of
mathematics – known for its abstract nature rather than its
applications. The great English mathematician, G.H. Hardy (1877 –
1947), used number theory as an example of a beautiful, but
impractical, branch of mathematics. However, in the late 1900s,
number theory became extremely useful in cryptosystems – systems
used for secure communications.
Find the...
Discrete Mathematics Probability Worksheet Name
__________________________________________
(1) Two ordinary dice are rolled. Find the probability that
...
(a) ... the sum of the dice is 6, 7 or 8.
(b) ... the sum of the dice is 5 or at least one of the dice
shows a 5
.(c) ... the two dice match.
(2) A card is drawn from an ordinary deck of 52 cards. Find the
probability that the card is ...
(a) ... an ace or a...
discrete mathematics
1.
Show that if a | b and b | a, where a and b are
integers, then a = b or a = -b.
//Ex. 5, Page 208.
2.
Show that if a, b, c, and d are integers such that a |
c and b | d, then ab | cd.
//Ex. 6, Page 208.
3.
What are the quotient and remainder when
a) 44 is divided by 8?
b) 777 is divided by 21?
f)...
Find a system of recurrence relations for the number of n-digit quaternary sequences that contain an even number of 2’s and an odd number of 3’s. Define the initial conditions for the system. (A quaternary digit is either a 0, 1, 2 or 3)
Let S(n) be the number of subsets of {1,2,...,n} having the
following property: there are no three elements in the subset that
are consecutive integers. Find a recurrence for S(n) and explain in
words why S(n) satisfies this recurrence
Problem 3:
Find the equilibrium distribution for each transition
matrix.
a)
1/2 1/9 3/10
1/3 1/2 1/5
1/6 7/18 1/2
b)
2/5 0 3/4
0 2/3 1/4
3/5 1/3 0
Problem 4:
For either transition matrix in problem 3, find the other two
eigenvalues with corresponding eigenvectors.
Problem 3:
Find the equilibrium distribution for each transition
matrix.
a)
1/2 1/9 3/10
1/3 1/2 1/5
1/6 7/18 1/2
b)
2/5 0 3/4
0 2/3 1/4
3/5 1/3 0
Problem 4:
For either transition matrix in problem 3, find the other two
eigenvalues with corresponding eigenvectors.