Discrete Mathematics Answers

You have two machines A and B that, each, generate binary digits. On each machine, when the “run” button is pressed, it will generate a single binary digit. Machine A generates a 0 - 52% of the times and a 1 - 48% of the times. Machine B, however, has a memory slot that stores the latest bit generated. Machine B always starts by generating a 0 and storing this in the memory slot. Each time, after this initialization, machine B generates the new bit by checking the bit in the memory slot, and generates the new bit by flipping the bit 61% of times and overwrites the memory with this new bit. When machine B is turned off, the memory slot clears itself.

(a) On using the ’run’ feature 7 times on Machine A, what is the probability that the outcome has exactly five 0’s?

(b) What is the probability of each machine generating ’00110’ and ’1001’?

Each point on a straight line is colored either red or blue. Prove that we can

find three points of the same color such that one is the midpoint of the other two.

On each square of a 5 × 5 board, there is a spider. Due to a sudden tremor

each spider jumps to an adjacent square (two squares are adjacent if they share an edge).

After this happens, is it possible that there is still one spider in each square?

Write down the negation of the following statements and determine the truth value of the negation:

a) ∀𝑥 ∈ 𝑹, 𝑥 2 + 1 ≥ 2𝑥

b) ∀𝑥 ∈ 𝑹, (𝑦 ≠→ (𝑦 + 1)/𝑦 < 1

c) ∃𝑧 ∈ 𝒁, (𝑧 𝑖𝑠 𝑜𝑑𝑑) ∨ (𝑧 𝑖𝑠 𝑒𝑣𝑒𝑛)

d) ∃𝑛 ∈ 𝑵, (𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛) ∧ (√𝑛 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒)

Let D = {-5, -3, -1, 1, 3, 5}. Write the following statements using only negations, conjunctions and disjunctions:

a) ∃𝑥𝑃(𝑥)

b) ∀𝑥𝑃(𝑥)

c) ∀𝑥((𝑥 ≠ 1) → 𝑃(𝑥))

d) ∃𝑥((𝑥 ≥ 0) ∧ 𝑃(𝑥))

e) ∃𝑥(￢𝑃(𝑥)) ∧ ∀𝑥((𝑥 < 0) → 𝑃(𝑥)) 

Let 𝑄(𝑥, 𝑦) denote "𝑥 + 𝑦 = 𝑦“. What are the truth values of the quantifications ∃𝑦∀𝑥𝑄(𝑥, 𝑦) and ∀𝑥∃𝑦𝑄(𝑥, 𝑦) where the domain for all variables consists of all real numbers?

Show that A = B where A = {x│x² - 4x +4 =1}, B = {1,3}

List all strings over X = {0,1}  of length 2 or less.

Using a truth table, prove or disprove the following:

~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r