# Proving associativity is a structural property

• kathrynag
In summary: So, in summary, to prove that * is commutative being a structural property, we need to show that associativity of a binary operation is preserved under isomorphisms, and this can be done by assuming two arbitrary isomorphic binary structures with an isomorphism between them and using that and the associativity of * to show that the other binary operation, *, is also associative.
kathrynag

## Homework Statement

Give a proof for the operation * is commutative being a structural property.

## The Attempt at a Solution

* is commutative
I know this means that I have to show (a*b)*c=a*(b*c)
I'm not sure where to go now

What is the definition of *, and what do you mean by "structural property"?

kathrynag said:

## The Attempt at a Solution

* is commutative
I know this means that I have to show (a*b)*c=a*(b*c)

You're confusing "commutative" and "associative".

oops, I menat to say associative. So, is that the right first step?

There's no way to tell if you're going in the right direction until you answer Citran's question. What is the definition of "structural property"?

A structural property of a binary structure is one that must be shared by any isomorphic structure.

* is just any arbitrary operation

Okay, so you want to show that associativity of a binary operation is preserved under isomorphisms. That is, if $(S,\ \ast)$ and $(T,\ \star)$ are isomorphic binary structures, and $\ast$ is associative, then $\star$ is also associative. Is that right?

If so, then what you will want to do is start by letting $(S,\ \ast)$ and $(T,\ \star)$ be arbitrary isomorphic binary structures with an isomorphism $\varphi: S \rightarrow T$ between them. Then assume that $\ast$ is associative, and use that and the isomorphism to show that $\star$ is associative.

That makes a lot more sense to me.

## 1. What is associativity?

Associativity is a mathematical property that states that the grouping of operands in an operation does not affect the outcome. In other words, it doesn't matter how you group the numbers, the final result will be the same.

## 2. Why is proving associativity important?

Proving associativity is important because it is a fundamental property in mathematics that allows us to simplify and manipulate equations. It is also a key concept in abstract algebra and is used in various fields of science and engineering.

## 3. How do you prove associativity is a structural property?

To prove that associativity is a structural property, we must show that it holds true for all possible combinations of operands. This can be done through algebraic manipulation or by using a proof by induction, depending on the specific context.

## 4. Can you give an example of associativity in action?

One example of associativity is the addition of real numbers. For instance, (2 + 3) + 4 = 2 + (3 + 4) = 9. The grouping of the numbers does not affect the final result, which is 9.

## 5. How does associativity relate to other mathematical properties?

Associativity is closely related to the commutative and distributive properties. The commutative property states that the order of operands does not affect the outcome, while the distributive property allows us to expand an operation over a set of parentheses. All three properties work together to simplify and manipulate equations in mathematics.

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