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Question 4. the Cartesian product Z₂ Z5 of two ([k], [m]) and Given the triple < Z₂ XZ₁, , > defined in terms of sets of congruence classes, Z₂ and Z5, under operations ([1], [n]) = ([k+1], [m+n]) ([k], [m]) ([4], [n]) = ([kl], [mn]) (a) Prove that the first distributive law holds true. (b) Hence prove that is a ring. (c) Is it a commutative ring? Justify your answer.

Question 4. the Cartesian product Z₂ Z5 of two ([k], [m]) and Given the triple < Z₂ XZ₁, , > defined in terms of sets of congruence classes, Z₂ and Z5, under operations ([1], [n]) = ([k+1], [m+n]) ([k], [m]) ([4], [n]) = ([kl], [mn]) (a) Prove that the first distributive law holds true. (b) Hence prove that is a ring. (c) Is it a commutative ring? Justify your answer.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 4. Given the triple < Z₂ x Z₁, B, D defined in terms of the Cartesian product Z₂ Z5 of two sets of congruence classes, Z₂ and Zs, under operations ([k], [m])=([1], [n]) = ([k+l], [m+n]) and ([k], [m]) ([1], [n]) = ([kl], [mn]) (a) Prove that the first distributive law holds true. (b) Hence prove that <Z₂ X Z₁, B, > is a ring. (c) Is it a commutative ring? Justify your answer.
Transcribed Image Text:Question 4. Given the triple < Z₂ x Z₁, B, D defined in terms of the Cartesian product Z₂ Z5 of two sets of congruence classes, Z₂ and Zs, under operations ([k], [m])=([1], [n]) = ([k+l], [m+n]) and ([k], [m]) ([1], [n]) = ([kl], [mn]) (a) Prove that the first distributive law holds true. (b) Hence prove that <Z₂ X Z₁, B, > is a ring. (c) Is it a commutative ring? Justify your answer.
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