Discrete Mathematics Answers

Decide whether each of these integers is congruent to 3 modulo 7.
(a) 37
(b) 66
(c) -17
(d) -67

Suppose a and b are integers, a ≡ 11 (mod 19) and b ≡ 3 (mod 19). Find an integer c with 0≤c≤18 such that
(a)c≡13a(mod 19).
(b)c≡8b(mod 19).
(c)c≡a-b(mod 19).
(d)c≡7a+ 3b(mod 19).
(e)c≡2a^2+ 3b^2(mod 19).
(f)c≡a^3+ 4b^3(mod 19).

What time does a 24-hour clock read
(a) 100 hours after it reads 2:00?
(b) 45 hours before it reads 12:00?
(c) 168 hours after it reads 19:00?

Let A be a set, and let P(A) denote the power set of A. Prove that|A|<|P(A)|.
Hint: Proceed in two steps.
1. First show that|A| <= |P(A)|. Try defining the function g: A-> P(A) by g(a) ={a}, and verify that g is one-to-one.
2. Then show that we can't have |A|=|P(A)|. Assume not, i.e., suppose that in fact |A|=|P(A)|. Then there exists a bijection f: A->P(A). Let B={aϵA|a Ɇf(a)} ϵ P(A)

Since f is onto, there exists an a0ϵA such that f(a0) =B. How does this lead to a contradiction?

Show that if A, B, C, and D are sets with |A|=|B| and|C|=|D|, then |AxC|=|BxD|.

If A is an uncountable set and B is a countable set, must A-B be uncountable?

Show that if A and B are sets where A⊆B and A is uncountable, then B is uncountable.

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