MAT2358 - Exam 2 2021 - all on one page-1 PDF

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Exam 2 for Kristen Lew's Discrete Mathematics class....

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LIC AT E

MATH 2358: Discrete Mathematics I Exam 2

Name: Problem

Out of

NO TD

1

Score

UP

April 8, 2021

2 3 4 5

Total Notes:

100

DO

• Read and follow the instructions for each problem. Be sure to understand each problem carefully before starting to work on it. ASK ME QUESTIONS IF YOU’RE UNSURE ABOUT SOMETHING! • Feel free to include scratch work, but clearly indicate what you want to be your final proof. • You can write on the exam or on your own paper. PDFs of your exam/work must be uploaded to Canvas no later than 5 minutes after class ends.

EX AM

• Make sure that your final responses are sufficiently clear and sufficiently justified. You do not want to lose points because I can’t understand what you wrote or doubt the reasoning. • Keep your camera on throughout the duration of the exam.

Relax and good luck!

ICA TE

MATH 2358, Spring 2021, Exam 2

1. Suppose A, B, C are sets. Prove that if A ⊆ B, then A − C ⊆ B − C . 2. Consider the function f (x) =

jx k 2

from R to R.

PL

Recall that f (x) = ⌊x⌋ is the greatest integer function, whose image of x is the greatest integer less than or equal to x. jx k one-to-one? Explain why or why not. (a) Is the function f (x) = 2 jx k onto? Explain why or why not. (b) Is the function f (x) = 2

DU

3. (a) Given 5x ≡ 7(mod 8), find the inverse of 5 modulo 8, then find x. (b) Find the gcd of 1785 and 546 using the Euclidean algorithm.

NO

4. (a) Convert the binary number (110011000101101010)2 to base 16, using the following table.

(b) Write the binary number (10101010)2 in base 10. Show your work.

DO

(c) Write the number (17853)10 in base 16. Show your work. 5. Use the Principle of Mathematical Induction to prove that 3|n3 + 2n whenever n is a positive integer. (Tip: Recall that (k + 1)3 = k 3 + 3k 2 + 3k + 1.)

Extra Credit:

i n X X

EX AM

(a) Evaluate the sum:

j for n = 5. Write out all terms of the sum in your work.

i=1 j=1

(b) Prove by mathematical induction that

n X i X

1 j = n(n + 1)(n + 2) for all positive integers n. 6 i=1 j=1

1...