Discrete Mathematics Answers

Show that a subset of a countable set is also countable

(7) An employee joined a company in 2017 with a starting salary of 50000 USD. Every year this employee receives a raise of 1000 USD plus 5 percent of the salary of the previous year. Let Cn denote the employee's salary n years after 2017.
(a) Set up a recurrence relation for Cn.
(b) What will the salary of this employee be in 2025?
(c) Find an explicit formula for Cn.

Compute each of the double double sums below
(a)3∑(i=1) 2∑(j=2) (i-j)
(b)3∑(i=0) 2∑(j=0) (3i+2j)
(c)3∑(i=1) 2∑(j=0) j
(d) 2∑(i=0) 3∑(j=0) i^2 j^3

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