Discrete Mathematics Answers

~(pVq)→r

1.Determine for which integer values of n, 3n^3+2≤n^4 and prove your claim by mathematical induction.

2.

1. Use iteration, either forward or backward substitution, to solve the recurrence relation an=an−1−1 for any positive integer n, with initial condition, a0=1. Use mathematical induction to prove the solution you find is correct.

2. Determine the cardinality of each of the sets, A, B, and C, defined below, and prove the cardinality of any set that you claim is countably infinite.

A is the set of negative odd integers

B is the set of positive integers less than 1000

C is the set of positive rational numbers with numerator equal to 1.

3.Using the definition of "Big-O" determine if each of the following functions, f(x)=(xlogx)^2−4 and g(x)=5x^5 are O(x^4) and prove your claims.

Let p and q be the propositions

p : It is below freezing. (including negations).

q : It is snowing

Write these propositions using p and q and logical connectives

a) It is below freezing and snowing.

b) It is below freezing but not snowing.

c) It is not below freezing and it is not snowing.

d) It is either snowing or below freezing (or both).

e) If it is below freezing, it is also snowing.

f ) Either it is below freezing or it is snowing, but it is

not snowing if it is below freezing.

Find the inverse of 35 modulo 11 by using extended Euclidean Algorithm

step by step solution

Find the inverse of 55 modulo 7 by using extended Euclidean Algorithm

step by step solution

The English alphabet contains 21 consonants and 5 vowels. How many strings of five lowercase letters can be formed using the following constraints? Give two answers for each of the following - one where repetition is allowed in the string and one where repetition is not allowed.

(a) Only one vowel (placed anywhere)

(b) Maximum two consonants (placed anywhere)

(c) Starts with x, y or z

State and prove Pascal’s identity using the formula for "{n \\choose k}"

Consider all strings of length 12, consisting of all uppercase letters. Letters may be repeated. Please do not simplify your answers.

(a) How many such strings are there?

(b) How many such strings contain the word ”SCOOBY”?

(c) How many such strings contain neither the word ”SCOOBY” nor the word ”DAPHNE”?

How many solutions in non-negative integers are there to the equation x₁ + x₂+ x₃ + x₄ = 19