Problem 3: For the following problem state the assumption (what is given and what is it that you aregoing to prove)ti start the proof with if you select each of the a,b,c proving techniques:Problem: If x>5 and y is even then x*y +3 will not be divisible by 2.a) Direct Proof:b) Contrapositive Proof:c) Contradiction Proof:

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Answered: Problem 3: For the following problem…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) 2.) (a) In how many ways can a hand of 6 hearts be chosen from an ordinary deck? (b) In how many ways can a hand of 6 black cards be chosen from an ordinary deck?
Question 2 Write a formal proof for each of the following claims; some of them were used in class. In what follows, K, a, b are positive integers. In each part you may use the previous parts of the question, even if you did not solve it. a) If b+ 0 and a | b then a 1 such that a b then a K. Hint: Use proof by contradiction. That is, assume that both a b and a | K hold and then use Part a) to reach a contradiction.
I need help with this problem and an explanation for the solution described below (Discrete mathematics):
3. If you have a jar of 1000 red and blue jelly beans and you 4 1 know that P(Blue) and P(Red) = list several 5 mathematical observations you can make about the num- ber of beans in the jar.
Choose correct answer on each problem
For these proofs I know have to use "subsets" but not sure how I think the 2nd problem may involve a proof with cases. Prove A - B is equivalent to A n Bnot Prove Anot U Bnot is equivalent to Anot n Bnot Prove if A subset B, then AnBnot = null (I think do a proof by contrapositive but not sure how)
The following is a list of converse statement of modular arithmetic. Determine if any of the following results hold true. ACTIVITIES AND ASSIGNMENTS FOR WEEK 2 3 (0) If a x k = b x k mod n for any integer k, then (r) If a = b' mod n for any positive integer l, then a = b mod n. (p) If a² = b² mod n, then a = b mod n. (g) If a³ = b³ mod n, then a = b mod n. a = b mod n. (s) If 2ª = 2º mod n, then a = b mod n. (t) If 3ª = 3º mod n, then a = b mod n. 3 Again, remember that the above list from (a) to (t) are not all true. Some of them are false. Pick your favorite fact and do the followings:
Problem 5: Another student has started a proof of the proposition Vn e Z where n> 4, n! > 2", and has written their basis step (shown below). Is this a valid basis step? Explain why or why not. Also, how many base cases would be necessary (minimally) for a proof of this proposition? Proof : WTS: Vn e Z where n > 4, n! > 2" Step 1: Base Case WTS: P(5) is true . LHS: 5! = 120. RHS: 25 = 32. LHS > RHS. So P(5) is true . WTS: P(6) is true . LHS: 6! = 720. RHS: 26 = 64. LHS > RHS.So P(6) is true . WTS: P(7) is true . LHS: 7! = 5040. RHS: 27 = 128. LHS > RHS. So P(7) is true .
One of the following is Not true about Pascal's triangle: O It is symmetric vertically (L)gac) O (7,6) means row 7 and column 6 O Counting both rows and columns starts from "one" O Its sides end with number "1"
Put the steps of a proof for the following claim in the proper order: 4 + 8 + 12 + + 4n = 2n(n + 1) + 4k + 4(k + 1) = 2k (k + 1) +4(k + 1) 2 (k + 1) (k +2) • 4+8+ 12 + .. + 4k = 2k (k + 1). Assume 4 + 8 + 12 + v Let n = 1. Clearly, 4 = 2(1)(1+ 1). v WTS: 4 +8+ 12 + + 4k + 4(k + 1) = 2 (k + 1) ((k + 1) + 1)

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