Midterm1

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 1/11 Q1 0 Points The exam is closed book. No phones or other devices are allowed for the exam. You may NOT use communication tools to collaborate with other humans about the exam. This includes but is not limited to any Chat, Messenger, E-mail, etc. Anyone found sharing ideas, solutions, or communicating with another student during the exam period will earn an immediate grade of 0. You have to choose the following option. Otherwise, your submission will NOT be considered. “I understand the ground rules and agree to abide by them. I will not share ideas, solutions, or assist another student during this exam, nor will I seek assistance from another student or attempt to view their ideas or solutions.” Q2 3 Points Which of the following are subsets of the set defined below? There may be more than one correct choice. In order to get a point for this question, you have to correctly select all that apply. I understand the ground rules. A A = {1, 2, {3}, {1, 2}, {3, 1, 2}}

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 2/11 Q3 3 Points Consider the following sets: For example, . Which of the following statements are True? There may be more than one correct choice. In order to get a point for this question, you have to correctly select all that apply. Q4 {3} {1, 2} {2} 1 A = ( x , x , x , x , x ) : x ∈ {0, 1, 2} for i = 1, 2, ..., 5 { 1 2 3 4 5 i } B = ( x , x , x , x , x ) : x ∈ Z for i = 1, 2, ..., 5 ∧ x + x + x + x + x ≥ { 1 2 3 4 5 i 1 2 3 4 5 C = ( x , x , x , x , x ) : x ∈ Z for i = 1, 2, ..., 5 ∧ x x x x x = 1 { 1 2 3 4 5 i 1 2 3 4 5 } (1, 0, 2, 0, 1) ∈ A ∣ A ∣ = 3 5 ∣ C ∣ = 2 5 ∣ A ∪ C ∣ = 3 + 5 2 5 ∣ A ∩ B ∣ = 5

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 4/11 19 Q7 6 Points Suppose we are given distinct balls and distinct colors, and we want to color each ball by one of the given colors. Each color can be used as many times as we wish. For example, two balls can be colored with the same color. Answer the following questions: Q7.1 2 Points In how many ways can we color the balls? You must only write down your final answer without any extra terms. Your answer must only be a number without any mathematical symbols like multiplication or addition. For example, if your answer is , just write down in the blank part. 256 Q7.2 2 Points In how many ways can we color the balls such that the balls are colored by more than one color? (In other words, not all the 4 balls are colored with the same color) You must only write down your final answer without any extra terms. Your answer must only be a number without any mathematical symbols like multiplication or addition. For example, if your answer is , just write down in the blank part. 252 4 4 1 1 1 1

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 5/11 Q7.3 2 Points In how many ways can we color the balls such that the first ball is colored differently from the next two balls? You must only write down your final answer without any extra terms. Your answer must only be a number without any mathematical symbols like multiplication or addition. For example, if your answer is , just write down in the blank part. 36 Q8 5 Points A “palindromic integer” is an integer that remains the same when its digits are reversed. Q8.1 2 Points How many -digit palindromic positive integers are there which are odd? You must only write down your final answer without any extra terms. Your answer must only be a number without any mathematical symbols like multiplication or addition. For example, if your answer is , just write down in the blank part. 50 Q8.2 3 Points How many -digit palindromic positive integers are there which are even? 1 1 4 1 1 5

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 7/11 Consider the mathematical statements . Prove that is logically equivalent to . You have to prove it without using a true/false table, otherwise you won't get any points. You can either typeset your solution in the blank part or upload your solution. IMG_6507.HEIC.pdf  Download A , B , C ( A ∨ B ) → C ( A → C ) ∧ ( B → C ) 

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 8/11 Q13 5 Points Consider the set , and the subset . How many subsets are there which satisfy ? Only providing the final answer is NOT enough. You have to also prove (or explain) your answer. S = {1, 2, 3, ..., 100} A = {1, 2, 3, ..., 30} B ⊆ S ∣ B − A ∣ = 1

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 10/11 those from the word becoming there are 4 left, therefore, there are 6 different combinations. 4. In how many of the permutations of the 10 letters in "becomingly", does at least one of "be" and "comi" appear? There would be 9! + 7! -6! which comes out to 367,200 different combinations where at least one of "be" and "comi". This is due to the fact that | A ∪ B | = |A| + |B| - |A ∩ B|. No files uploaded  Graded Midterm 1 Student Devesh Jaiswal Total Points 62 / 72 pts Question 1 (no title) 0 / 0 pts Question 2 (no title) 3 / 3 pts Question 3 (no title) 3 / 3 pts Question 4 (no title) 3 / 3 pts Question 5 (no title) 3 / 3 pts Question 6 

7/21/23, 4:55 PM View Submission | Gradescope https://www.gradescope.com/courses/145902/assignments/564863/submissions/40241591#Question_13-rubric 11/11 (no title) 3 / 3 pts Question 7 (no title) 4 / 6 pts 7.1 (no title) 2 / 2 pts 7.2 (no title) 2 / 2 pts 7.3 (no title) 0 / 2 pts Question 8 (no title) 5 / 5 pts 8.1 (no title) 2 / 2 pts 8.2 (no title) 3 / 3 pts Question 9 (no title) 0 / 3 pts Question 10 (no title) 0 / 3 pts Question 11 (no title) 3 / 3 pts Question 12 (no title) 11 / 11 pts Question 13 (no title) 3 / 5 pts + 5 pts Correct  + 3 pts Partial incorrect answer + 4 pts missing parts in the proof. + 2 pts incorrect answer + 0 pts no answer  The correct answer is 70*(2^30) Question 14 (no title) 21 / 21 pts