Let S be the set S = {a+bk : a,b ∈ R}, where k is a formal symbol. Define addition and multiplication operations on S as follows: given elements x = a+bk and y = c+dk in S, x+y := (a+c) + (b+d)k, xy := (ac+bd) + (ad +bc)k. Prove that the multiplicative inverse law is false for S. [That is, don’t just write down a counterexample; also prove that your counterexample is valid.]

Let S be the set S = {a+bk : a,b ∈ R}, where k is a formal symbol. Define addition and multiplication operations on S as follows: given elements x = a+bk and y = c+dk in S, x+y := (a+c) + (b+d)k, xy := (ac+bd) + (ad +bc)k. Prove that the multiplicative inverse law is false for S. [That is, don’t just write down a counterexample; also prove that your counterexample is valid.]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

20200 ecc 7. Given sets A, B and C prove or disprove that B) U (B-C) = (A-C)_
4. Prove that, if f : A → B has an inverse, then f is bijective. (The converse is also true, and very useful, but you do not need to prove it.) If A is a set, a binary operation on A is a function 3: Ax A→ A. Binary operations are ubiquitous in modern algebra, and their appearance there motivates the following notation: for (a1, a2) € A × A, we write a₁ßa2 instead of the usual functional notation B(a₁, a2) for the image of (a₁, a2) under 3. The most important (and motivating) instances of this (very general) notion for us are the addition and multiplication operations on a ring.
In the set C of ordered pairs of real numbers, addition and multiplication are defined by the formulae (a, b)+(c, d)=(a + c, b +d) and (a, b).(c, d)=(ac - bd , ad + bc). Show that the system (C,+, .) is a field. What are the additive and multiplicative identities of of this field ?
Define the binary operator ⋆⋆ by: a⋆b=a+8ba⋆b=a+8b Find each of the following: 3⋆4=3⋆4= 9⋆9=9⋆9= 4⋆3=4⋆3= z⋆c=z⋆c=
Let Zg = {0,1,2,3,4,5,6,7}. Denote by xg the operation "multiplication modulo 8" where a xg b is the remainder when a × b is divided by 8. 3a) The identity element of Zg under X8 is 3c) Determine the inverse of each of the following elements of Zg with respect to Xg. Element of Z Inverse with respect to X8 3 5 7
Find out whether the set of integers Z with respect to addition ± and multiplication O is defined as follows:
On R we define the following binary operation; x*y = xy + 2ax + by Find a and b such that * is commutative and associative.
Show using tableaux that the following wff is valid:(p → q) → (p → (q ∨ r))You may either eliminate implications or use your above rules for them. However, using other identities to simplify your wffs (other than for implications) does not qualify as a tableau proof