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Abstract Algebra
Consider the following sets together with binary operations.
Are they user-friendly? Z with binary operation
z1 . z2 = 2z1 - 4z2
Is the set closed under the operation? Is the operation commutative ?
Is the operation associative ? Is there an identity? If there is an identity
element then does every element have an inverse relative to the operation _
Consider R together with x . y = x/y. Ask the same questions as in last
example.
Are they user-friendly? Z with binary operation
z1 . z2 = 2z1 - 4z2
Is the set closed under the operation? Is the operation commutative ?
Is the operation associative ? Is there an identity? If there is an identity
element then does every element have an inverse relative to the operation _
Consider R together with x . y = x/y. Ask the same questions as in last
example.
Abstract Algebra
Prove or disprove that C~ R as fields.
Abstract Algebra
1. \\((p\\wedge q)= (q\\wedge p)\\) and \\((p\\vee q) = (q\\vee p)\\) implies an _____ a. Idempotent Laws b. Associative laws c. Distributive Laws d. Commutative Laws 2. given that\\(A=\\begin{pmatrix}1 & 2 & 3\\\\ 4 & 5 & 6 \\end{pmatrix}\\)\nand \\[B=\\begin{pmatrix} 1 & 2\\\\ 3& 4\\\\ 5& 6 \\end{pmatrix}\\]\n. Find AB a. \\(\\begin{pmatrix} 0 & 12 & 17\\\\ 19 & 26 & 31\\\\ 29 & 40 & 51 \\end{pmatrix}\\) b. \\(\\begin{pmatrix} 5 & 12 & 15\\\\ 19 & 26 & 31\\\\ 29 & 40 & 51 \\end{pmatrix}\\) c. \\(\\begin{pmatrix} 0 & 12 & 17\\\\ 19 & 26 & 31\\\\ 20 & 40 & 45 \\end{pmatrix}\\) d. \\(\\begin{pmatrix} 0 & 12 & 17\\\\ 7 & 10 & 31\\\\ 20 & 40 & 45 \\end{pmatrix}\\)
Abstract Algebra
If p*q = p^2-q^2-2pq. Find the inverse of p under the operation.
Abstract Algebra
State and prove generalized commutative law in a commutative semigroup
Abstract Algebra
Q. Find the dimension of the subspace of R4 that is span of the vectors
(█(1¦(-1)@0@1)), (█(2¦1@1@1)),(█(0¦0@0@0)),(█(1¦1@-2@-5))
Q. Choose the correct answer.
Q. Let b and c are elements in a group G and e is identity element of G. If b5=c3=e,then inverse of bcb2 is
a. b2cb
b. b3c2b4
d. b2c2b4
(█(1¦(-1)@0@1)), (█(2¦1@1@1)),(█(0¦0@0@0)),(█(1¦1@-2@-5))
Q. Choose the correct answer.
Q. Let b and c are elements in a group G and e is identity element of G. If b5=c3=e,then inverse of bcb2 is
a. b2cb
b. b3c2b4
d. b2c2b4
Abstract Algebra
A linear equation over R can have at most one root in C\R.
Is the statement true or false? Justify your answer.
Is the statement true or false? Justify your answer.
Abstract Algebra
a) Let d∈N , where d ≠1 and d is not divisible by the square of a prime.
Prove that N:Z[√d]→N∪{0}:N(a+b√d)=|a²-db²| satisfies the following properties for x,y∈Z[√d] :
i) N(x) = 0⇔x=0
ii) N(xy) = N(x)N(y)
iii) N(x) =1⇔ x is a unit
iv) N(x) is prime ⇒x is irreducible in Z[√d ] .
b) Prove or disprove that C≃ R as fields.
Prove that N:Z[√d]→N∪{0}:N(a+b√d)=|a²-db²| satisfies the following properties for x,y∈Z[√d] :
i) N(x) = 0⇔x=0
ii) N(xy) = N(x)N(y)
iii) N(x) =1⇔ x is a unit
iv) N(x) is prime ⇒x is irreducible in Z[√d ] .
b) Prove or disprove that C≃ R as fields.
Abstract Algebra
a) If G is a group of order 40, and H and K are its subgroups of orders 20 and 10,
then check whether or not HK ≤ G . Further, show that o(H∩K) ≥ 5.
b) Prove that C*/S~=R+, where S ={z∈C*| |z|=1}, R+ ={x∈R| x>0} and C*=C\{0}.
c) What are the possible algebraic structures of a group of order 99?
then check whether or not HK ≤ G . Further, show that o(H∩K) ≥ 5.
b) Prove that C*/S~=R+, where S ={z∈C*| |z|=1}, R+ ={x∈R| x>0} and C*=C\{0}.
c) What are the possible algebraic structures of a group of order 99?
Abstract Algebra
If ω be the imaginary cube root of unity show that the set {1,w,w^2)is a cyclic group of order 3 with respect to multiplication.
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#340153 on Dec 2023
#340153 on Dec 2023