Search & Filtering
Let D=(3,5,7,9) E=( 0, 4, 6, 9) F=0,3,6,7 . Universal set U = 0,1,2,3,4,5,6,7,8,9.
Draw venn diagram for.
(I) DnEnF
(II) (DUE) Uf
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”?
Refer to a group of 191 students, of which 10 are taking math, business, and language; 36 are taking math and business; 20 are taking math and language; 18 are taking business and language; 65 are taking math; 76 are taking business and 63 are taking language.
26-30. Illustrate the Venn Diagram
Prove or disprove that if R and S are antisymmetric, then so is:
(a) (R ∪ S)
(b) (R ∩ S)
In Monopoly, your token is allowed to leave the ”jail” cell if you roll doubles: you roll two 6 -sided dice and each shows the same face. Zach hates being in jail, So he invents a couple of weighted dice that are not independent. In particular, if you roll either die on its own it’s a fair die: each outcome has probability 1/6. But if you roll one die and then the other, the red die will take the same outcome as the blue die exactly half the time: all other outcomes are equally likely.
(a) Suppose you roll a 3 on the blue die. What is the probability distribution of the red die given this outcome on the blue die?
(b) What is the probability you roll doubles?
(c) What is the probability that you roll a 7 as the sum of the two dice?
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))
