5) Determine whether the given definition does give a binary operation on the indicated set. In...
5, b) Determine whether the definition of * does give a binary operation on the set and give reason why a-b On R define * by letting a * b a k IV
Consider the following examples of a set S and a binary operation on S. Show with proof that the binary operation is indeed a binary operation, whether the binary operation has an identity, whether each element has an inverse, and whether the binary operation is associative. Hence, determine whether the set S is a group under the given binary operation. (f) S quadratic residues in Z101 under multiplication modulo 101 Consider the following examples of a set S and a...
1. Determine whether * is a binary operation on the given set. If it is a binary operation, decide whether it is associative and commutative. Justify your answers. a. Define * on Q+ by a *b = b. Define * on N by a*b = %.
The set G = {a ∈ Q| a≠0} is closed under the binary operation a ∗ b = ab/3 . Prove that (G, ∗) is an abelian group. 4. (10 points) The set G = {a e Qla #0} is closed under the binary operation a*b = ab 3 Prove that (G, *) is an abelian group.
Which of the following sets, together with the given binary operation *, DOES NOT form a group? (Notation: As usual, the notations Z, Q, R, and C represent the sets of integers, rational numbers, real numbers, and complex numbers, respectively.) (A.) G is {a+bV2 ER\{0} | a, b e Q}; * is the usual multiplication of real numbers (B.) G is {a + biv2 € C\{0} | a, b E Q}; * is the usual multiplication of complex numbers (C.)...
Modern Algebra 5) Consider the ollowing sets, S, together with the defined binary operation. In each case, determine if the set is closed under the given operation, if the operation is associative and if the operation is commutative: ii) S R a -a b 6) Define the binary operation, multiplication modulo 3 in much the same way as we did addition modulo 3. That is, perform ordinary multiplication and then reduce the result modulo 3. Let S-(0, 1,2. Create two...
Question 2. Recall that a monoid is a set M together with a binary op- eration (r,y) →エ. y from M × M to M, and a unit element e E/, such that: . the operation is associative: for all x, y, z E M we have (z-y): z = the unit element satisfies the left identity axiom: for all r E M we have the unit element satisfies the right identity axiom: for all a EM we Let K...
(10 points) The set G = {a e Qla #0} is closed under the binary operation a * b ab 3 Prove that (G, *) is an abelian group.
Determine if the set V = {at? | a € R} is a subspace of the vector space P2 = {ao +ajt + azt? | ao, a1, az ER}. You may assume that vector addition in P2 is given by the usual addition of polynomials and that the scalars used in scalar multiplication are real numbers. If you decide that Vis a subspace of P2, then identify the zero vector in V and explain briefly why Vis closed under vector...
linear algebra 1. Determine whether the given set, along with the specified operations of addition and scalar multiplication, is a vector space (over R). If it is not, list all of the axioms that fail to hold. a The set of all vectors in R2 of the form , with the usual vector addition and scalar multiplication b) R2 with the usual scalar multiplication but addition defined by 31+21 y1 y2 c) The set of all positive real numbers, with...