Multiplication Properties of Equivalence Classes

In summary, the conversation discusses the proof or disproof of the statement "for [a], [b] ∈ Zn for a positive integer n, if [a]·[b]=[0], then either [a]=[0] or [b]=[0]." The statement is disproved by providing a counterexample, and a new statement is proposed as a possible salvage: "For [a], [b] ∈ Zn for a positive integer n, if [a] · [b] = [0], then ab ≡ 0 (mod n)." The conversation then discusses how to prove this new statement.
  • #1
jcfaul01
2
0

Homework Statement



Prove or disprove and salvage if possible: for [a], ∈ Zn for a positive integer n, if [a]·=[0], then either [a]=[0] or =[0].

The Attempt at a Solution



I've managed to disprove the statement:
Let n=6,[a]=3,and=[4]. The[a]·=[ab]=[3·4]=[12]. Since12≡0(modn), 12 ∈ [0] so[12] = [0]. Thus [3] · [4] = [0] and this statement is false.

However, my problem is with salvaging it. I've been able to come up with what I believe to be the correct statement:

For [a], ∈ Zn for a positive integer n, if [a] · = [0], then ab ≡ 0 (mod n).

But I have no idea how to prove it, I don't even know where to start. I would really appreciate any help on this,

thanks.
 
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  • #2
What you have, [m][n]= 0 implies mn= 0 (mod n) is just the definition of [m][n]= 0.

Let n be a prime integer, then ...
 
  • #3
Thanks! I think I've got it now!
 

What are the Multiplication Properties of Equivalence Classes?

The Multiplication Properties of Equivalence Classes refer to the rules and properties that govern the multiplication operation on equivalence classes. These properties are important in understanding and solving mathematical problems involving equivalence classes.

What is the Commutative Property of Multiplication for Equivalence Classes?

The Commutative Property of Multiplication for Equivalence Classes states that the order of multiplication does not affect the result. In other words, for any two equivalence classes A and B, A * B = B * A. This property allows us to change the order of multiplication without changing the result.

What is the Associative Property of Multiplication for Equivalence Classes?

The Associative Property of Multiplication for Equivalence Classes states that the grouping of three or more equivalence classes does not affect the result. In other words, for any three equivalence classes A, B, and C, (A * B) * C = A * (B * C). This property allows us to change the grouping of multiplication without changing the result.

What is the Identity Property of Multiplication for Equivalence Classes?

The Identity Property of Multiplication for Equivalence Classes states that the existence of an identity element in the set of equivalence classes. In other words, there exists an equivalence class I, such that A * I = A and I * A = A for any equivalence class A. This property ensures that every equivalence class has a "neutral" element under multiplication.

What is the Zero Property of Multiplication for Equivalence Classes?

The Zero Property of Multiplication for Equivalence Classes states that the existence of a zero element in the set of equivalence classes. In other words, there exists an equivalence class 0, such that A * 0 = 0 and 0 * A = 0 for any equivalence class A. This property ensures that any equivalence class multiplied by the zero element will result in the zero element.

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