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Discrete Mathematics
February 2025
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Discrete Mathematics
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February 12, 2025
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February 12 of 2025
18. Let \( \mathcal{U}=\{1,2,3\} \) be the universal set for \( A=\{1,2\} \) and \( B=\{2,3\} \). (a) \( \mathcal{P}(A) \cap \mathcal{P}(B) \), (b) \( \mathcal{P}(\bar{A}) \cup \mathcal{P}(\bar{B}) \), (c) \( \mathcal{P}(A)-\mathcal{P}(B) \).
13. Let \( A, B, C \) be sets with \( A \neq \emptyset \). Prove that if \( A \times B=A \times C \), then \( B=C \). Is the statement still true if \( A=\emptyset \) ? Prove your answer.
50% of Jada's shirts have short sleeves. If she has 20 short-sleeved shirts, how many shirts does Jada have altogether?
Suppose a student’s final grade in a biology course is determined using the following weights: Quizzes are worth 5% Exam 1 is worth 20% Exam 2 is worth 20% Lab reports are worth 15% Research paper is worth 15% Final exam is worth 25% Just before the final, she has earned the following…
6. [1 mark] \( A B \) is parallel to \( C D \). Work out the size of angle \( y \). Give reasons for your answer.
Exercise 3.4: A small auditorium labels its seats with one of the 26 a-z letters and one of the ten digits \( 0-9 \). How many different seat labels are possible? What is the answer if each seat is labeled with one of the letters a-z and one of the number 1-100? What is the answer if each seat…
Exercise 3.8: At a CI, students are given a 9 digit student ID number. How many different ID numbers are there? How many read the same forward and backward? How many contain only odd digits? How many have at least on even digit?
Exercise 3.3: How many strings of 2 lowercase letters have the letter \( x \) in them? How many strings of 4 lowercase letters have the letter \( x \) in them? How many strings of \( k \) lowercase letters have the letter \( x \) in them?
an express bus left jalandhar at 18.25 hours and reaches delhi at 6.25 hours.how long time did the journey take?
Use induction to prove: \[ 1+5+9+\ldots+4 n-3=n(2 n-1) \]
5. \( 8 \frac{1}{3}-3 \frac{4}{5}= \) a. \( 5 \frac{7}{15} \) b. \( 5 \frac{2}{15} \) c. \( 4 \frac{8}{15} \) d. \( 4 \frac{4}{15} \) 6. A door opening is 68 inches high, and Jason is \( 62 \frac{1}{2} \) inches tall. How much taller is the door frame than Jason? a. \( 4 \frac{1}{2} \)…
The pie chart shows how 36 pupils travel to school. Use the pie chart to complete the table. \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Travel to \\ school \end{tabular} & \begin{tabular}{c} Number \\ of pupils \end{tabular} \\ \hline Walk & \\ \hline Bus & \\ \hline Car & \\ \hline…
(1) The joint probability density function of the two random variables and \( Y \) is \( f(x, y)=8 x y, 0 \leq x \leq y \leq 1 \). Find \( f_{Y x}\left(y \left\lvert\, \frac{1}{3}\right.\right) \) \( x=3 \)
2.7. Suppose \( X \) is a random variable having the probability density function \[ f(x)=\left\{\begin{array}{ll} R x^{R-1} & \text { for } 0 \leq x \leq 1 \\ 0 & \text { elsewhere } \end{array}\right. \] where \( R>0 \) is a fixed parameter. (a) Determine the distribution function \( F_{x}(x)…
\[ F(x)=\left\{\begin{array}{ll} 0 & \text { for } x \leq 0 \\ x^{3} & \text { for } 0<x<1 \\ 1 & \text { for } x \geq 1 \end{array}\right. \] b) Determine the corresponding density function \( f(x) \) in the three regions (i) \( x \leq 0 \), (ii) \( 0<x<1 \), and (iii) \( 1 \leq x \). c) What…
5. The number of accidents occurring in a factory in a week js a Poisson random variable with mean 2. The number of individuals injured in different accidents is independently distributed, each with mean 3 and variance 4 . Determine the mean and variance of the number of individuals injured in…
3. Find the orthogonal trajectories to the family of curves \( y=\mathrm{Cx}^{3} \), and geve a rough skedel of both families of curves.
Use induction to prove: \[ 1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}=\frac{4 n^{3}-n}{3} \]
\( \begin{array}{r}5.72 \\ \times 1.1\end{array} \)
Use induction to prove: \[ 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{n(n+1)(n+2)}{3} \]
The baseband signal is m(t) = A1 cos(2πf1t) + A2 cos(2πf2t) (1) where A1 = 1, A2 = 0.8, f1 = 100Hz and f2 = 200Hz. The carrier signal is c(t) = Ac cos(2πfct) (2) where Ac = 2 and fc = 10KHz. The AM signal is: s(t) = Ac(1 + Kam(t)) cos(2πfct) (3) And the double sideband suppressed…