Discrete

Problem 1

1. Prove $n^{7}-n$ is divisible by 42 for all positive integers $n$ . Show primes ≠ 2, 5 divide numbers like 1, 11, etc.
2. Prove if $p>3$ is prime, then $p^{2} \equiv 1(\bmod 24)$ .
3. Find the number of trailing zeros in $1000!$ .
4. If $p$ and $p^{2}+2$ are primes, prove $p^{3}+2$ is prime.
5. Prove $\operatorname{gcd}(2^{a}-1,2^{b}-1)=2^{\operatorname{gcd}(a, b)}-1$ for positive integers $a, b$ .

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Problem 2

Tom has 4 different marbles and 3 identical yellow ones. How many unique pairs of marbles can he select?

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Problem 3

Tom has 4 unique marbles and 3 identical yellow ones. How many ways can he choose 2 marbles?

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Problem 4

Ibrahim dan Zahra datang ke mesjid setiap 4 dan 6 hari. Apakah mereka bersama lagi pada 12, 14, dan 26 Juli 2022?

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Problem 5

Ibrahim dan Zahra datang bersama pada 2 Juli 2022. Kapan mereka akan bersama lagi? Tandai dengan $\sqrt{ }$ pilihanmu!
1. 12 Juli 2022
2. 14 Juli 2022
3. 26 Juli 2022
4. 7 Agustus 2022

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Problem 6

How many unique 4-letter passwords can be formed from the 26-letter English alphabet without repeating letters? [?] ways

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Problem 7

How many ways can you choose 3 books from 11? Use the formula for combinations: $\binom{11}{3}$ .

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Problem 8

What is the converse of "If I studied for it, then I did well on the exam"?

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Problem 9

Calculate $12 / 20$ and find the missing number in the sequence: 100, 93, 84, 73.

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Problem 10

Find the next number in the sequence: $6, 3, 12, 9, 36, 33$ .

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Problem 11

If this animal is a dog, it has four legs. Which statement is equivalent? A, B, or C?

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Problem 12

How many ways can you play 3 songs from 8 using random shuffle? Use ${ }_{8} \mathrm{P}_{3}=\frac{8 !}{(8-3) !}$ .

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Problem 13

Choose 3 light bulbs from a batch of 100. How many ways can this be done? Use ${ }_{100} \mathrm{C}_{3}=\frac{100 !}{3 !(100-3) !}$ .

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Problem 14

En un grupo de 100 personas, 60 hablan inglés, 50 alemán y 15 no hablan ninguno. ¿Cuántas hablan ambos idiomas?

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Problem 15

Sean A y B subconjuntos de un conjunto Re. ¿Cuáles son las afirmaciones verdaderas sobre ellos?

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Problem 16

Identify the FALSE statement among these:
a) $A(p(x) \rightarrow q(x))=A^{c} q(x) \cup A p(x)$
b) $A(p(x) \vee q(x))=A p(x) \cup A q(x)$
c) $A(p(x) \wedge q(x))=A p(x) \cap A q(x)$
d) $A^{c} q(x)=A \neg q(x)$
e) $(q(a) \equiv 1) \rightarrow(a \in A q(x))$

See Solution

Problem 17

Identify which sequence is arithmetic and which is geometric: Sequence #1: $5, 15, 45, 135, 405$ ; Sequence #2: $5, 15, 25, 35, 45$ .

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Problem 18

Find the 6th and 7th terms of the sequence $7n-5$ ; also, find the sum of the 100th and 200th terms. Is 99 a term?

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Problem 19

En la secuencia, cada número es la suma de los dos anteriores. ¿Cuál es el valor de $x$ en la fila? 10, 16, 28, 26, 149.

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Problem 20

Un vector tiene magnitud y dirección, mientras que un escalar solo tiene magnitud. ¿Cuál es la diferencia?

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Problem 21

For $n \geq 100$ , show that in two piles of cards numbered $n$ to $2n$ , at least one pile has two cards summing to a perfect square.

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Problem 22

What is the probability of randomly selecting the correct answer: A: $25\%$ , B: $0\%$ , C: $50\%$ , D: $25\%$ ?

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Problem 23

Show that $p \Rightarrow(q \wedge r) \equiv(p \Rightarrow q) \wedge(p \Rightarrow r)$ is always true.

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Problem 24

In a solid 'AB' with NaCl structure, if one face's atoms are removed, what is the new stoichiometry? (1) A₄B₃ (2) AB (3) A₃B₄ (4) AB₂

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Problem 25

How many ways can you choose 5 numbers from 1 to 59 and 1 bonus number from 1 to 35?

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Problem 26

How many ways can 10 athletes finish in the top 3 positions of a $400 \mathrm{~m}$ race? Options: a. 720 b. 120 c. $10^{3}$ d. 3

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Problem 27

A college conference has 3 presenters in Hall A, 2 in Hall B, and 4 in Hall C. How many ways to attend 2 from C and 1 from A or B?

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Problem 28

How many ways can you attend 2 presentations in Hall C and 1 in either Hall A or B if there are 3 in A, 2 in B, and 4 in C?

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Problem 29

Prove the following for subsets $A, B \subset S$ : (a) $A \subset B$ iff $A \cup B = B$ ; (b) $A \subset B$ iff $A \cap B = A$ ; (c) $A \subset C(B)$ iff $A \cap B = \varnothing$ ; (d) $C(A) \subset B$ iff $A \cup B = S$ ; (e) $A \subset B$ iff $C(B) \subset C(A)$ ; (f) $A \subset C(B)$ iff $B \subset C(A)$ .

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Problem 30

Find the union of the sets $A=\{x \mid x$ is a prime number between 1 and 10 $\}$ and $B=\{x \mid x$ is an even number between 1 and 10 $\}$ .

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Problem 31

Given the sets $A=\{1,2,3,4,5,6,7,8,9,10\}$ and $B=\{2,4,6,8,10,12,14,16,18,20\}$ , find $A \cap B'$ .

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Problem 32

Find the intersection of set A and the complement of set B, where A = {1,2,...,10} and B = {2,4,...,20}.

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Problem 33

Find the union of set $A$ and the intersection of sets $B$ and $C$ : $A \cup (B \cap C)$ .

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Problem 34

Toss a fair coin and a four-sided die. Find the probability of heads and an even die result. Complete the diagram.

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Problem 35

Execute $n$ processes with given times. When executing time[i], reduce all equal times to $\lceil \text{time}[i]/2 \rceil$ . Find total execution time.

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Problem 36

Goldbach's conjecture states that every even number greater than 2 can be written as a sum of two primes. Is it true? Why?

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Problem 37

Find the next number in the sequence: $3,4,1,8,5,2,9,10,1, ?$ a. 13 b. 6 c. 5 d. 7

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Problem 38

Identify the pattern in the sequence $0, 3, 18, 57$ and extend it. Prove your conclusion using induction.

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Problem 39

Find the intersection of set $A$ and the complement of set $B$ within the universe $U$ .

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Problem 40

Find the union of set A and the intersection of sets B and C: $A \cup (B \cap C)$ .

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Problem 41

Find the union of the sets $A=\{x \mid x$ is a prime number between 1 and 10 $\}$ and $B=\{x \mid x$ is an even number between 1 and 10 $\}$ .

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Problem 42

एक व्यक्ति को 3 और 5 लीटर के बर्तनों से बिना अन्य बर्तन का उपयोग किए 4 लीटर पानी बनाना है। क्या वह कर सकता है?

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Problem 43

False

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Problem 44

Is the statement true or false? $9 \in\{2,4,6, \ldots, 20\}$ True or False?

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Problem 45

Is the statement true or false? $17 \notin\{1,2,3, \ldots, 10\}$

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Problem 46

Is the statement true or false? If false, explain why. $4 \in\{1,3,5,7, \ldots\}$

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Problem 47

Find the union of sets $A=\{a, b, c, d\}$ and $B=\{b, c, 1, 2\}$ . What is $A \cup B$ ? A) $\{a, b, 1, 2\}$ B) $\{a, b, c, d, 1, 2\}$ C) $\{d, c, a, a, 2\}$ D) $\{a, b, c, 1, 2\}$

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Problem 48

Find the Cartesian products: $A \times B$ , $B \times A$ , and $A \times A$ for $A=\{a,b,c\}$ and $B=\{1,2\}$ .

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Problem 49

Decida si 2 es un elemento de cada conjunto: a. $\{\{\{2\}\}\}$ , b. $\{2,\{2\}\}$ , c. $\{x \in \mathbb{R} \mid x$ es el cuadrado de un entero $\}$ , d. $\{x \in \mathbb{R} \mid x$ es un número mayor que 1 $\}$ , e. $\{\{2,\{2\}\}\}$ , f. $\{\{2\},\{2,\{2\}\}\}$ .

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Problem 50

a.) ¿Cuántas delegaciones diferentes se pueden formar con al menos un miembro de 18 estudiantes? b.) ¿Cuántas delegaciones diferentes se pueden formar con al menos dos miembros de 18 estudiantes?

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Problem 51

Is the set $\{a, b, c\}$ an element of the set $\{a, b, c, \{a, b, c\}\}$ ?

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Problem 52

Choose $\in$ , $\subseteq$ , or their negations to complete: $\phi$ $\_\_\_$ $\{x ; x$ is a negative number between 20 and 21 $\}$ . Options: a. $\notin$ , b. $\subseteq$ , c. $\in$ , d. $\nsubseteq$ .

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Problem 53

Encuentra y simplifica la intersección de $\{-\infty, 4] \cap(0, \infty)$ .

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Problem 54

หาจำนวนรหัส 4 ตัวที่สร้างจากพยัญชนะ ก, ข, ค, ง โดยไม่ซ้ำกัน (24)

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Problem 55

11. หาเลข 3 หลักคู่ที่ใช้เลขซ้ำและไม่ซ้ำได้ทั้งหมด $(450,328)$
12. หาวิธีนั่ง 3 คู่สามีภรรยา โดยผู้ชายต้องนั่งปลายแถวเสมอ (144)
13. จัดหลอดไฟสีแดง 3 หลอด, เหลือง 4 หลอด, น้ำเงิน 2 หลอด ใน 9 ขั้ว (1,260)
14. จัดแถว 5 คนจากห้อง ก, 3 คนจากห้อง ข, 2 คนจากห้อง ค โดยนักเรียนห้องเดียวกันติดกัน (8,640)

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Problem 56

หาเลข 3 หลักคู่ที่มีเลขซ้ำและไม่ซ้ำกัน $(450,328)$ และวิธีนั่งของสามีภรรยาที่ผู้ชายต้องนั่งปลายแถว (144)

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Problem 57

Rappresenta con un diagramma di Eulero-Venn gli insiemi: a. $A=\{x \mid x$ è un giorno della settimana $\}$ b. $B=\{x \mid x$ è un mese che inizia con $A\}$ c. $C=\{x \mid x$ è una cifra di 96.820 $\}$

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Problem 58

Given sets $X=\{1,2,3\}$ and $Y=\{13,31,65,23\}$ , do the following:
1. Write the relation $R$ and draw an arrow diagram.
2. Determine if $2 R 65$ , $1 R 31$ , $3 R 13$ are true, with reasons.
3. List $M=\{(y, 23):(y, 23) \in R\}$ .

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Problem 59

Describe the set $B$ where $B=\{1,2\}$ .

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Problem 60

Are the sets $A=\{\text{literature, geometry, algebra}\}$ and $B=\{\text{geometry, trigonometry, literature}\}$ equal, equivalent, both, or neither?

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Problem 61

Check if sets $A$ (letters in 'dime') and $B$ (letters in 'car') are equal, equivalent, both, or neither.

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Problem 62

Find the number of terms in the sequence $20, 18, 16, \ldots$ that ends with -4.

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Problem 63

In a survey of 100 traders selling fruits, find those selling only oranges and mangoes using a Venn diagram.

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Problem 64

Use the OPT replacement algorithm on the page references $0,6,3,0,2,6,3,5,2,4,1,3,0,6,1,4,2,3,5,7$ . How many page faults with 3 frames?

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Problem 65

A 32-bit address with 2KB page size requires how many entries in a single-level page table?

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Problem 66

Arrange 8 students with the following conditions: 1) A & B together: 10,080; 2) A & B at ends: 1,440; 3) A & B apart: 30,240; 4) Alternating genders: 1,152; 5) 4 boys & 4 girls together: 1,152.

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Problem 67

Identify the domain and range of the set $\{(3,2),(5,7),(1,4),(9,2),(3,7)\}$ and determine if it is a function.

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Problem 68

Find the domain and range of the set $\{(1,2),(2,5),(3,1),(1,6),(4,8)\}$ . Is it a function?

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Problem 69

Determine the domain and range of the set $\{(6,2),(3,5),(9,0),(5,7),(8,1)\}$ and check if it's a function.

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Problem 70

Find the domain and range of the set of points $\{(1,9),(2,7),(5,4),(7,12),(3,9)\}$ .

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Problem 71

Find the domain and range of the set $\{(0,2),(3,3),(8,7),(2,2),(3,9)\}$ and determine if it's a function.

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Problem 72

Find the page number and offset for the virtual address 30000 with a page size of 2048 Bytes.

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Problem 73

A web server tracks hits with multi-threading. Is the lock method valid for preventing race conditions? Options: A) True B) Mutex C) None D) Not valid due to concurrency E) Not valid due to non-atomic increase.

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Problem 74

Given six memory partitions: A (100 MB), B (170 MB), C (40 MB), D (205 MB), E (300 MB), F (185 MB) and processes P1 (200 MB), P2 (15 MB), P3 (185 MB), P4 (75 MB), P5 (175 MB), P6 (80 MB). Use worst-fit to find remaining memory in D.

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Problem 75

Select all possible outputs of the given C program when run on a Unix system. Possible outputs include:
1. ABBDD
2. A A B C D
3. A B D C D
4. ACBDD
5. ABBCCDD
6. ACD
7. A B C D
8. A B C D D

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Problem 76

Calculate the average waiting time for processes P1 (12), P2 (4), P3 (8), P4 (2), and P5 (6) using Shortest-Job-First scheduling.

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Problem 77

Identify possible outputs of the given C program when run on a Unix system. Options include: ABBDD, A A B C D, A B D C D, ACBDD, ABBCCDD, ACD, A B C D, A B C D D.

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Problem 78

A web server tracks request hits using a lock. Is this solution valid for preventing race conditions? Choose true statements.

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Problem 79

Find the next term in the sequence: 2, 2, 4, 6, 10, 16, 26, 42. Provide your answer as an integer, decimal, or E notation.

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Problem 80

Find the next term in the sequence: 45, 94, 143, 192.

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Problem 81

Find the next term in the sequence: 55, 66, 78, 91.

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Problem 82

Match the sequences: $126,139,152,165,178$ ; $89,222.5,556.25,1390.625,3,476.5625$ ; $3,3,6,9,15,24$ ; $36,45,55,66,78$ ; $441,484,529,576,625$ with sequence types: Fibonacci, Arithmetic, Cube, Square, Geometric, Triangular.

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Problem 83

An office has 8 floors. Which set represents the $x$ -coordinates for the elevator stops? Options: A) {1,2,3,4,5,6,7,8}, B) {0,1,2,3,4,5,6,7}, C) {0,1,2,3,4,5,6,7,8}, D) {-1,-2,-3,-4,-5,-6,-7,-8}.

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Problem 84

Soit $I$ et $J$ deux intervalles ouverts, avec $(I \cap \mathbb{Q}) \cap (J \cap \mathbb{Q}) = \varnothing$ . Prouvez que $I \cap J = \varnothing$ .

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Problem 85

Identify the correct notation for permutation from these options: $\mathrm{rPn}$ , $P(n, r)$ , $C(n, r)$ , $\mathrm{nCr}$ .

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Problem 86

Найдите $n(B)$ , если $n(A \cup B)=31$ , $n(A)=18$ , $n(A \cap B)=9$ .

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Problem 87

Who can always manage a close corporation? Options: a. (4) b. (1), (2) & (4) c. (1) & (4) d. (1), (2), (3) & (4) e. (2) & (4)

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Problem 88

Find the total number of members in three teams: cricket (21), hockey (26), football (29), with overlaps given.

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Problem 89

Identify the correct concepts related to negation and matrices, and find the power set of the empty set.

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Problem 90

Find the next number in the series: $36, 34, 30, 28, 24, \ldots$ .

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Problem 91

Determine if the statement "Gamestation $\in\{$ Port'n'Play, Gamestation, Port'n'Play EX $\}$ " is true or false. If false, explain why.

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Problem 92

Determine if $\{ Larry, Fred \} \subseteq \{ Curly, Larry, Moe \}$ is true or false. If false, explain why.

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Problem 93

Determine if the statement "7 is not in the set {1, 5, 6}" is true or false. If false, explain why.

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Problem 94

Determine the relationship between sets $A=\{$ winter, autumn, summer, spring $\}$ and $B=\{$ winter, spring $\}$ .

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Problem 95

Find an index $i$ in array $\mathrm{A}$ where the sum before $i$ equals the sum after $i$ . Return $-1$ if none exist.

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Problem 96

What is the probability of guessing a 5-digit code from 5 unique numbers? A. $\frac{1}{12}$ B. $\frac{1}{120}$ C. $\frac{1}{40}$ D. $\frac{1}{60}$

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Problem 97

How many elements are in the union of sets A and B if A has 8 elements, B has 20, and 5 are common?

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Problem 98

In a pet supply store survey, 27 owned dogs, 30 owned mice, and 17 owned both. Find how many owned either a dog or a mouse.

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Problem 99

A restaurant gives $10\%$ of customers a discount coupon. Which model simulates this? A) Spinner, B) Dice, C) Cards, D) Random numbers?

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Problem 100

What is the next number in the sequence? $1, 2, 2, 4, 8, 32, ?$

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119

Discrete

Problem 1

Problem 2

Tom has 4 different marbles and 3 identical yellow ones. How many unique pairs of marbles can he select?

Problem 3

Tom has 4 unique marbles and 3 identical yellow ones. How many ways can he choose 2 marbles?

Problem 4

Ibrahim dan Zahra datang ke mesjid setiap 4 dan 6 hari. Apakah mereka bersama lagi pada 12, 14, dan 26 Juli 2022?

Problem 5

Ibrahim dan Zahra datang bersama pada 2 Juli 2022. Kapan mereka akan bersama lagi? Tandai dengan \sqrt{ }​ pilihanmu! 1. 12 Juli 2022 2. 14 Juli 2022 3. 26 Juli 2022 4. 7 Agustus 2022

Problem 6

How many unique 4-letter passwords can be formed from the 26-letter English alphabet without repeating letters? [?] ways

Problem 7

How many ways can you choose 3 books from 11? Use the formula for combinations: (113)\binom{11}{3}(311​).

Problem 8

What is the converse of "If I studied for it, then I did well on the exam"?

Problem 9

Calculate 12/2012 / 2012/20 and find the missing number in the sequence: 100, 93, 84, 73.

Problem 10

Find the next number in the sequence: 6,3,12,9,36,336, 3, 12, 9, 36, 336,3,12,9,36,33.

Problem 11

If this animal is a dog, it has four legs. Which statement is equivalent? A, B, or C?

Problem 12

How many ways can you play 3 songs from 8 using random shuffle? Use 8P3=8!(8−3)!{ }_{8} \mathrm{P}_{3}=\frac{8 !}{(8-3) !}8​P3​=(8−3)!8!​.

Problem 13

Choose 3 light bulbs from a batch of 100. How many ways can this be done? Use 100C3=100!3!(100−3)!{ }_{100} \mathrm{C}_{3}=\frac{100 !}{3 !(100-3) !}100​C3​=3!(100−3)!100!​.

Problem 14

En un grupo de 100 personas, 60 hablan inglés, 50 alemán y 15 no hablan ninguno. ¿Cuántas hablan ambos idiomas?

Problem 15

Sean A y B subconjuntos de un conjunto Re. ¿Cuáles son las afirmaciones verdaderas sobre ellos?

Problem 16

Problem 17

Identify which sequence is arithmetic and which is geometric: Sequence #1: 5,15,45,135,4055, 15, 45, 135, 4055,15,45,135,405; Sequence #2: 5,15,25,35,455, 15, 25, 35, 455,15,25,35,45.

Problem 18

Find the 6th and 7th terms of the sequence 7n−57n-57n−5; also, find the sum of the 100th and 200th terms. Is 99 a term?

Problem 19

En la secuencia, cada número es la suma de los dos anteriores. ¿Cuál es el valor de xxx en la fila? 10, 16, 28, 26, 149.

Problem 20

Un vector tiene magnitud y dirección, mientras que un escalar solo tiene magnitud. ¿Cuál es la diferencia?

Problem 21

For n≥100n \geq 100n≥100, show that in two piles of cards numbered nnn to 2n2n2n, at least one pile has two cards summing to a perfect square.

Problem 22

What is the probability of randomly selecting the correct answer: A: 25%25\%25%, B: 0%0\%0%, C: 50%50\%50%, D: 25%25\%25%?

Problem 23

Show that p⇒(q∧r)≡(p⇒q)∧(p⇒r)p \Rightarrow(q \wedge r) \equiv(p \Rightarrow q) \wedge(p \Rightarrow r)p⇒(q∧r)≡(p⇒q)∧(p⇒r) is always true.

Problem 24

In a solid 'AB' with NaCl structure, if one face's atoms are removed, what is the new stoichiometry? (1) A₄B₃ (2) AB (3) A₃B₄ (4) AB₂

Problem 25

How many ways can you choose 5 numbers from 1 to 59 and 1 bonus number from 1 to 35?

Problem 26

How many ways can 10 athletes finish in the top 3 positions of a 400 m400 \mathrm{~m}400 m race? Options: a. 720 b. 120 c. 10310^{3}103 d. 3

Problem 27

A college conference has 3 presenters in Hall A, 2 in Hall B, and 4 in Hall C. How many ways to attend 2 from C and 1 from A or B?

Problem 28

How many ways can you attend 2 presentations in Hall C and 1 in either Hall A or B if there are 3 in A, 2 in B, and 4 in C?

Problem 29

Problem 30

Find the union of the sets A={x∣xA=\{x \mid xA={x∣x is a prime number between 1 and 10}\}} and B={x∣xB=\{x \mid xB={x∣x is an even number between 1 and 10}\}}.

Problem 31

Given the sets A={1,2,3,4,5,6,7,8,9,10}A=\{1,2,3,4,5,6,7,8,9,10\}A={1,2,3,4,5,6,7,8,9,10} and B={2,4,6,8,10,12,14,16,18,20}B=\{2,4,6,8,10,12,14,16,18,20\}B={2,4,6,8,10,12,14,16,18,20}, find A∩B′A \cap B'A∩B′.

Problem 32

Find the intersection of set A and the complement of set B, where A = {1,2,...,10} and B = {2,4,...,20}.

Problem 33

Find the union of set AAA and the intersection of sets BBB and CCC: A∪(B∩C)A \cup (B \cap C)A∪(B∩C).

Problem 34

Toss a fair coin and a four-sided die. Find the probability of heads and an even die result. Complete the diagram.

Problem 35

Execute nnn processes with given times. When executing time[i], reduce all equal times to ⌈time[i]/2⌉\lceil \text{time}[i]/2 \rceil⌈time[i]/2⌉. Find total execution time.

Problem 36

Goldbach's conjecture states that every even number greater than 2 can be written as a sum of two primes. Is it true? Why?

Problem 37

Find the next number in the sequence: 3,4,1,8,5,2,9,10,1,?3,4,1,8,5,2,9,10,1, ?3,4,1,8,5,2,9,10,1,? a. 13 b. 6 c. 5 d. 7

Problem 38

Identify the pattern in the sequence 0,3,18,570, 3, 18, 570,3,18,57 and extend it. Prove your conclusion using induction.

Problem 39

Find the intersection of set AAA and the complement of set BBB within the universe UUU.

Problem 40

Find the union of set A and the intersection of sets B and C: A∪(B∩C)A \cup (B \cap C)A∪(B∩C).

Problem 41

Find the union of the sets A={x∣xA=\{x \mid xA={x∣x is a prime number between 1 and 10}\}} and B={x∣xB=\{x \mid xB={x∣x is an even number between 1 and 10}\}}.

Ibrahim dan Zahra datang bersama pada 2 Juli 2022. Kapan mereka akan bersama lagi? Tandai dengan $\sqrt{ }$ pilihanmu!
1. 12 Juli 2022
2. 14 Juli 2022
3. 26 Juli 2022
4. 7 Agustus 2022

How many ways can you choose 3 books from 11? Use the formula for combinations: $\binom{11}{3}$ .

Calculate $12 / 20$ and find the missing number in the sequence: 100, 93, 84, 73.

Find the next number in the sequence: $6, 3, 12, 9, 36, 33$ .

How many ways can you play 3 songs from 8 using random shuffle? Use ${ }_{8} \mathrm{P}_{3}=\frac{8 !}{(8-3) !}$ .

Choose 3 light bulbs from a batch of 100. How many ways can this be done? Use ${ }_{100} \mathrm{C}_{3}=\frac{100 !}{3 !(100-3) !}$ .

Identify which sequence is arithmetic and which is geometric: Sequence #1: $5, 15, 45, 135, 405$ ; Sequence #2: $5, 15, 25, 35, 45$ .

Find the 6th and 7th terms of the sequence $7n-5$ ; also, find the sum of the 100th and 200th terms. Is 99 a term?

En la secuencia, cada número es la suma de los dos anteriores. ¿Cuál es el valor de $x$ en la fila? 10, 16, 28, 26, 149.

For $n \geq 100$ , show that in two piles of cards numbered $n$ to $2n$ , at least one pile has two cards summing to a perfect square.

What is the probability of randomly selecting the correct answer: A: $25\%$ , B: $0\%$ , C: $50\%$ , D: $25\%$ ?

Show that $p \Rightarrow(q \wedge r) \equiv(p \Rightarrow q) \wedge(p \Rightarrow r)$ is always true.

How many ways can 10 athletes finish in the top 3 positions of a $400 \mathrm{~m}$ race? Options: a. 720 b. 120 c. $10^{3}$ d. 3

Find the union of the sets $A=\{x \mid x$ is a prime number between 1 and 10 $\}$ and $B=\{x \mid x$ is an even number between 1 and 10 $\}$ .

Given the sets $A=\{1,2,3,4,5,6,7,8,9,10\}$ and $B=\{2,4,6,8,10,12,14,16,18,20\}$ , find $A \cap B'$ .

Find the union of set $A$ and the intersection of sets $B$ and $C$ : $A \cup (B \cap C)$ .

Execute $n$ processes with given times. When executing time[i], reduce all equal times to $\lceil \text{time}[i]/2 \rceil$ . Find total execution time.

Find the next number in the sequence: $3,4,1,8,5,2,9,10,1, ?$ a. 13 b. 6 c. 5 d. 7

Identify the pattern in the sequence $0, 3, 18, 57$ and extend it. Prove your conclusion using induction.

Find the intersection of set $A$ and the complement of set $B$ within the universe $U$ .

Find the union of set A and the intersection of sets B and C: $A \cup (B \cap C)$ .

Find the union of the sets $A=\{x \mid x$ is a prime number between 1 and 10 $\}$ and $B=\{x \mid x$ is an even number between 1 and 10 $\}$ .

Is the statement true or false? $9 \in\{2,4,6, \ldots, 20\}$ True or False?

Is the statement true or false? $17 \notin\{1,2,3, \ldots, 10\}$

Is the statement true or false? If false, explain why. $4 \in\{1,3,5,7, \ldots\}$

Find the Cartesian products: $A \times B$ , $B \times A$ , and $A \times A$ for $A=\{a,b,c\}$ and $B=\{1,2\}$ .

Is the set $\{a, b, c\}$ an element of the set $\{a, b, c, \{a, b, c\}\}$ ?

Choose $\in$ , $\subseteq$ , or their negations to complete: $\phi$ $\_\_\_$ $\{x ; x$ is a negative number between 20 and 21 $\}$ . Options: a. $\notin$ , b. $\subseteq$ , c. $\in$ , d. $\nsubseteq$ .

Encuentra y simplifica la intersección de $\{-\infty, 4] \cap(0, \infty)$ .

หาเลข 3 หลักคู่ที่มีเลขซ้ำและไม่ซ้ำกัน $(450,328)$ และวิธีนั่งของสามีภรรยาที่ผู้ชายต้องนั่งปลายแถว (144)

Rappresenta con un diagramma di Eulero-Venn gli insiemi: a. $A=\{x \mid x$ è un giorno della settimana $\}$ b. $B=\{x \mid x$ è un mese che inizia con $A\}$ c. $C=\{x \mid x$ è una cifra di 96.820 $\}$

Describe the set $B$ where $B=\{1,2\}$ .

Are the sets $A=\{\text{literature, geometry, algebra}\}$ and $B=\{\text{geometry, trigonometry, literature}\}$ equal, equivalent, both, or neither?

Check if sets $A$ (letters in 'dime') and $B$ (letters in 'car') are equal, equivalent, both, or neither.

Find the number of terms in the sequence $20, 18, 16, \ldots$ that ends with -4.

Use the OPT replacement algorithm on the page references $0,6,3,0,2,6,3,5,2,4,1,3,0,6,1,4,2,3,5,7$ . How many page faults with 3 frames?

Identify the domain and range of the set $\{(3,2),(5,7),(1,4),(9,2),(3,7)\}$ and determine if it is a function.

Find the domain and range of the set $\{(1,2),(2,5),(3,1),(1,6),(4,8)\}$ . Is it a function?

Determine the domain and range of the set $\{(6,2),(3,5),(9,0),(5,7),(8,1)\}$ and check if it's a function.

Find the domain and range of the set of points $\{(1,9),(2,7),(5,4),(7,12),(3,9)\}$ .

Find the domain and range of the set $\{(0,2),(3,3),(8,7),(2,2),(3,9)\}$ and determine if it's a function.