Given a set A = {0,1, z} equipped with addition + and multiplication · defined by the tables: + 0 1 0 1 z 00 1 1 1 0 0 0 0 1 0 1 z 1 z 1 Answer the following questions: 1. Is multiplication commutative? [ Select) 2. Is there an additive identity? (Select] 3. Is there a multiplicative identity? [ Select ] >

Q: determine whether multiplication by A is one-to- one by computing the nullity of A and then applying…

A: Let (x, y) be in nullspace of A.  Then, A(x, y)t = 1-22-4-36xy = 000     => x - 2y = 0…

Q: In the ring Z[x]/ 1- The inverse of x is: a) x+1 b) x2 +1 c) x d) x? 2- The value that replacesx is…

A:

Q: Verify that for all x, the change-of-base formula holds: bx = ax Ioga b for a, b > 0 such that a…

A:

Q: Let {0,1}N = {(ao, a1, a2, ...) : an = 0 or 1, n E N}. • Construct a 1-1 correspondence between {0,…

A: Let          0, 1ℕ=(a0, a1, a2, . . . ) : an=0 or 1, n∈ℕ.

Q: Let 2 = {a, b, c}. Write down all possible o-algebras on N.

A: Given: Ω=a,b,c

Q: Let S be the set S = {a+bk : a,b ∈ R}, where k is a formal symbol. Define addition and…

A: Let xy=ac+bd+ad+bck Then first we find multiplicative identity. Let x=a+bky=c+dk Then…

Q: 3. If x is uncountable, then prove that A = {ECX: ECX: algebra O X: E is countable or E² is…

A:

Q: 1. For every element x in A, there is an element y in B such that (x, y) E F.

A:

Q: Suppose that E is an extension of F and a, b E E. If a is algebraic over F of degree m, and b is…

A:

Q: Let L be a Lie algebra and let C(L) be the centre of L. Show that an equality dim L/C(L) = 1 is…

A: The center of a lie algebra L is defined as CL=x∈L:x,y=0 for all y∈L. x,y=0⇒x and y commute. That is…

Q: Let K' be and K' be to divisions of [a,b]. Show that  L(f,K') ≤ U(f,K'')

A: We need to prove the following : If f is a bounded function on [a, b], and if P and Q are partitions…

Q: Prove that if x is even and y is even, then the product of x and y is divisible by 4.

A: We use the Division Algorithm to prove the given statement.

Q: Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

A: I have proved using the fact that gcd can be expressed as integers linear combination q

Q: The class of all finite disjoint unions of intervals of the form (a, b), (-0o, b), (b, o0), 0, and R…

A: Given: The class of all finite disjoint union of intervals of the form (a,b] ,(-∞,b] ,(b ,∞) ,∅ and…

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: The objective is to find Z_2×Z_2.

Q: Prove the de nitions of addition and multiplication in Zm [x]m + [y]m = [x + y]m and [x]m[y]m =…

A:

Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

Q: Question 9. (a) Prove that every element of Q/Z has finite order.

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: (1) Since (1)(1) - (0)() = 1- 0 = | Hence IE G La ea be any element a b b. Now identity so be the…

A: Follow step 2nd .

Q: Given an n x n-matriz A over a endent iff det(A) = det(A¹,...,A") =

A:

Q: Let a, b € Z and both nonzero. Prove that gcd(a, b) x lcm(a, b) = |ab|.

A:

Q: 1. For every element x in A, there is an element y in B such that (x, y)EF

A: Remark: Let A and B be two non-empty sets. A relation R from A into B is subset of A*B. A function f…

Q: f is said to be a if it is both one to one and onto. That is, there is a one to one correspondence…

A: f if said to be a------- if it is both one to one and onto.

Q: Prove that all elements in S are multiples of 3. by Structualindrction 3ES X,Y ES > (X y )ES

A:

Q: If |G|=77, then Z(G) is isomorphic to Z11 0 Zj1 O Z1 0 Z7 O Z7 O Z7 0 None of them Z77 0

A: Given that G = 77  then we have to find Z(G)   We first factorize 77 .

Q: Show that for every x E R, there exists an n EN such that <x<n.

A:

Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: Prove that every element of Sn (n &gt; 1) can be written as a productof elements of the form (1k).

A:

Q: Prove: Provided it exists, (A-1)¯ = A.

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: (3) Show that (C SRICs countable or C is antable) is a -algebra on R.

A:

Q: Given a set Q× Q under operations: (21, Y1) + (#2, Y2) = (¤1 + #2, Y1 + y2), (21,41) (22, Y2) =…

A: Given a set ℚ×ℚ under operations:(x1,y1)+(x2,y2)=(x1+x2, y1+y2)(x1,y1).(x2,y2)=(x1x2, y1y2)

Q: Let G = Z, O Z, H = {(0, 0), (2, 0), (0, 2), (2, 2)}, and K = ((1, 2)). Is GIH isomorphic to Z, or…

A:

Q: s] Assume that is an operation on S with identity element e, and that x * (y * 2) = (x * z2) * y for…

A: A operation  " o " on S is said to be commutative if x o y = y o x  for all x and y in S  And…

Q: 10. Let a, be K be algebraic over F of degrees m and n, respectively. If m and n are relatively…

A:

Q: Suppose that E is an extension of F, and a,b are elements of E. If a is algebraic over F of degree…

A:

Q: 1. Let X and Y be non empty sets and let £: XY be given. Prove that if M is a algebra of subsets of…

A:

Q: Prove: [a] is an element of Zn has a multiplicative inverse if and only if gcd(a,n)=

A:

Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions

Q: Suppose  there exist transposition f_i , 1<=I<=t  such that

A:

Q: The identity element of binary operation defined on ℝ by x * y= xy  is

A:

Q: Let a ∈ R[[x]] with a0 not equal to 0. Prove that a has a multiplicative inverse in R[[x]]. You may…

A: We have given that , a∈ℝx a = a0 + a1x + a2x2 + ......  where each ai ∈ ℝ. Since , 1 ∈ ℝx  is an…

Q: For a given set S, let P(S) be the collection of all subsets of S. Let binary operations + and · on…

A:

Q: Prove, by comparing orders of elements, that Z8 ⨁ Z2 is not isomorphicto Z4 ⨁ Z4.

A: Given  Z8 ⨁ Z2 and Z4 ⨁ Z4.

Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: For a given set S, let P(S) be the collection of all subsets of S. Let binary operations + and · on…

A: Given algebraic structure forms a commutative ring, but it is not an integral domain and hence it is…

Q: Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).

A: We have gcd(-a , b)=gcd(a,-b) let gcd(-a , b)=d Where -a=de  , b=df  ⇒a=d(-e) , -b=d(-f)⇒d is…

Q: a) Let * be the operation on Z defined by a*b= a + b - 3ab for all a,b e Z. Justify the following…

A:

Q: 6. Consider the subring S = {[0], [2], [4], [6], [8], [10], [12], [14], [16] } C Z18 with respect to…

A: The answer is given as follows:

Q: In each part following, a rule that determines a binary operation * on the set Z of all integers is…

A: The given operation is x*y=xy for all x,y ∈Z+