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Show that the set A = { 1,2,4,5,7,8 }with an operation as multiplication modulo 9 is a cyclic group.
Find the order of various elements and subgroup generated by them.
Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation
defined by aOb = max{ a,b}.
Prove or disprove: Each of the following is a ring. Determine their group of
units in each case.
The set of Gaussian integers modulo n for n ∈ N, i.e. Zn[i] = {[a + bi] | [a], [b] ∈ Zn}
together with operations ⊕ and , and i = √−1.
For [a1], [b1], [a2], [b2] ∈Zn, then
[a1 + b1i] ⊕[a2 + b2i] = [a1 + a2] ⊕[b1 + b2]i
and
[a1 + b1i] [a2 + b2i] = [a1a2 −b1b2] ⊕[a1b2 −b1a2]i
Prove/disprove that the set of all continuous functions C[0, 1] defined from the closed unit interval into R,
together with function addition + and function multiplication · is a ring
Prove for any ring R and a,b∈ R , (a+b)²= a²+2ab+b²
The subtraction of a matrix B may be considered as the addition of the marix (-1)B.Does the cummutative law of addition permit us to state that A-B=B-A?If not,how would you correct the statement?
let R be an equivalence relation and assume c R a and c R b.prove a R b
Check whether R is a group under binary operation
a*b=a+b-ab
Prove that Z27 is not a homomorphic image of Z72
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#340153 on Dec 2023