EXERCISES1.Consider the following operations on Za)m*n=m2 + n²b)m*n (m+n)/2Prove or disprove:1.a) and b) are binary operations on Z2.a) and b) are associative operations3.a) and b) are commutative operations4.a) and b) have identity elements with respect to *CL 39and2.Define.on R the operation o by a o b = a+ b- ab, wheremeaning in R. Prove:ia binary oneration on R

Let us prove or disprove the properties 1- 4 in the next steps.

Answered: 1. Consider the following operations on…

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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Let * be any binary operation on the set of real numbers defined by a * b=a + b – ab, then the binary operation * is ___.
4. Prove the following statement using a direct proof. Explicitly indicate the assumptions (what you suppose) and the conclusion (what you must show) of the proof. If b is divisible by a and c is divisible by b, where a, b, and c are integers with a + 0 and b + 0, then c is divisible by a.
which of the following is a binary operation if a and b are in S
Proficiency #7. [Counterexamples] For each of the following two statements, give a counterexample to show that the state- ment is false. a. If x is an integer such that x? = 1 mod 12, then x =1 mod 12 or x = -1 mod 12. b. If A, B, D are sets such that A C B and B Z D, then A Z D.
a. Does the definition of A give a binary operation on the set Zt of positive integers, where a Ab=b – 3 is the definition of A? b. Is the binary operation * on the set Zt of positive integers associative, where a * b = a + b is the definition of *? c. Let * be a binary operation on the set {0, 1}, which is defined by the operation table: * 1 0 0 1 10 Is the binary operation * commutative on the set A? Does the binary operation * have an identiy element on the set A? Is the binary operation * an idempotent operation on the set A? ?
Prove that the additive and multiplicative inverses of each nonzero elements in R are unique note that R is a set of real numbers
Please help with q36, 35 and 34
(a) Which of the numbers 0, 1, 2, . . . , 10 have (multiplicative) inverses mod 26?(b) For the numbers in (a) which have inverses mod 26, compute the inverses.
Answer in detail
5 Prove: ACB iff AxAC BxB
5. Let a and b be positive integers such that gcd(a, b+1) = 1 and b2 = 1 (mod a).What is the remainder in the division of b by a? Justify your answer.
D. Define the multiplication operation * as modulo 10 in G = {1, 3, 5, 7,}. Create the table for this. Verify the theorem (a * -b) = b-1 * a. 1. (7 * 5)-1 2. 5-1 * 7-1

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1. Consider the following operations on Z a) m*n = m2 + n? b) m*n (m+n)/2 Prove or disprove: 1. a) and b) are binary operations on Z 2. a) and b) are associative operations 3. a) and b) are commutative operations 4. a) and b) have identity elements with respect to *

1. Consider the following operations on Z a) mn = m2 + n? b) mn (m+n)/2 Prove or disprove: 1. a) and b) are binary operations on Z 2. a) and b) are associative operations 3. a) and b) are commutative operations 4. a) and b) have identity elements with respect to *