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Abstract Algebra Answers

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Question #94720
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}\\)


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Question #94719
Abstract Algebra
1. ____is equivalent to \\((p\\vee q)\\)
a. \\(\\sim (\\sim p\\wedge \\sim q)\\)
b. \\((\\sim \\sim q)\\)
c. \\((p\\leftrightarrow q)\\)
d. \\((\\sim p\\wedge q)\\)

2. \\((p\\vee q)\\vee r=p\\vee (q\\vee r)\\) and \\((p\\wedge q)\\wedge r = p\\wedge (q\\wedge r)\\) implies an ___
a. Distributive Laws
b. Commutative Laws
c. Associative laws
d. Idempotent Laws
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Question #94509
Abstract Algebra
Let d ∈N , where d ≠1 and d is not divisible by the square of a prime. Prove that N:Z[square root of d] maps N union {0} : N(a+b sqr root d) = |a^2 -db^2| satisfies the following properties for x,y belongs to Z[sqr root d]. 1. N(x) = 0 if x=0. 2 N(xy) = N(x)N(y) 3. N(x) =1 if x is a unit 4. N(x) is prime if x is irreducible.
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Question #93911
Abstract Algebra
If p*q = p^2-q^2-2pq. Find the inverse of p under the operation.
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Question #92042
Abstract Algebra
State and prove generalized commutative law in a commutative semigroup
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Question #91473
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
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Question #89033
Abstract Algebra
Let N ∈d , where d ≠1 and d is not divisible by the square of a prime. Prove that N:Z[square root of d] maps N union {0} : N(a+b sqr root d) = |a^2 -db^2| satisfies the following properties for x,y belongs to Z[sqr root d].

1. N(x) = 0 if x=0.
2 N(xy) = N(x)N(y)
3. N(x) =1 if x is a unit
4. N(x) is prime if x is irreducible.
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Question #88935
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.
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Question #87265
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.
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Question #87227
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.
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