Discrete Mathematics Answers

What are the values of these sums, where S={1,3,5,7}?
(a)∑(jϵS) j
(b)∑(jϵS) j^2
(c)∑(jϵS) (1/j)
(d)∑(jϵS) 1

Compute the values of the sums below.
(a)5∑(k=1) (k+ 1)
(b)4∑(j=0) (-2) ^j
(c) 10∑(i=1) 3
(d) 8∑(j=0) ( 2^(j+1) -2^j )

In class we showed the following: n ∑(k=1) k = n(n+1)/2 and n∑(k=1) k^2 = n(n+ 1)(2n+ 1)/6
Using the fact that (k+ 1)^4-k^4= 4k^3+ 6k^2+ 4k+ 1 and summing up as k= 1,2,3,……, n together with the above two equalities, deduce that n∑(k=1) k^3 = (n(n+1)/2)^2

Let ρ be a relation on a set A. Define ρ^−1 = {(b, a) | (a, b) ∈ ρ}. Also, for two relations ρ, σ on
A, define the composite relation ρ ◦ σ as (a, c) ∈ ρ ◦ σ if and only if there exists b ∈ A such that (a, b) ∈ ρ and (b, c) ∈ σ. Prove the following assertions.

i.) If ρ is non-empty, then ρ is an equivalence relation if and only if ρ^−1 ◦ ρ = ρ.

Let ρ be a relation on a set A. Define ρ^−1 = {(b, a) | (a, b) ∈ ρ}. Also, for two relations ρ, σ on
A, define the composite relation ρ ◦ σ as (a, c) ∈ ρ ◦ σ if and only if there exists b ∈ A such that (a, b) ∈ ρ and (b, c) ∈ σ. Prove the following assertions.

(i) ρ is a partial order if and only if ρ^−1 is a partial order

Prove or disprove that there exists a bijection from (0, 1] to [0, ∞)^2.

Prove or disprove that there exists a bijection from (0, 1] to (0, 1]^2

Suppose that the number of bacteria in a colony triples every hour. Let Bn denote the number of bacteria in the colony after n hours.
(a) Set up a recurrence relation for Bn.
(b) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in 10 hours?

1) A person deposits 1000 USD into an account that yields 9 percent interest compounded annually. Let An denote the amount of money in the account after n years.
(a) Set up a recurrence relation for An.
(b) Find an explicit formula for An.
(c) How much money will be in the account after 100 years?

Let U = {English, French, History, Math, Physics, Chemistry, Psychology, Drama},

A = {English, Chemistry, French, Psychology},



B = {Math, Physics, History, French, Psychology},

and

C = {Drama, Chemistry, History}.

Find the following.

n[(A ∪ B) ∪ C]