Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [associativity]

This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$.

This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$. This is a key property of groups, rings and fields.

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What is the mistake with this proof that commutativity implies associativity?

I tried to show that commutativity of addition implies associativity. For this I assumed that there is no associative property and $ a + b + c $ should be interpreted as $(a + b) + c $ . $$(a + b) + c = a + b + c = b + a + c = c + b + a = (c + b)…
Martx
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Why is the associative property so special to mathematicians?

A few years back I came across an article on quantum physics in Quanta Magazine. It described the work of Cohl Furey trying to plumb the secrets of the universe using octonions. The article explains: Much more bizarrely, the octonions are…
Cort Ammon
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Finding an associative operator with two operands, from its behaviour on three operands

Suppose I have forgotten how to calculate $g(a,b)=a+b$. But I have a black-box function $f(a,b,c)=a+b+c$, which I can calculate. Presumably, $f(a,b,c)=g(a,g(b,c))=g(g(a,b),c)$ uniquely specifies $g$ (does it?). Naturally, $g(a,b)\equiv…
Wouter
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Commutativity + "something" $ \to $ associativity?

What additional properties must an operation have besides commutativity so that commutativity along with other properties implies associativity? Where can I read about such structures?
user480281
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example of non-associativity in the physical universe?

In a recent article[1], John Baez is quoted as making a nice point about how non-commutativity is common in the world around us, whereas non-associativity is not: “[...] while it’s very easy to imagine noncommutative situations—putting on shoes…
Ruben Verresen
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How to correctly do a division using a slash?

What's the correct way to do division using a slash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$? Basically, if I place a slash, will I divide by what's directly behind the slash $(b)$, the term that's behind the slash $(bc)$, or…
Jeroen
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why is this associative?

I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already The fourth exercise asks me to determine whether the following operation is compliant with the associative principle: $$(α, β) · (γ, δ) = (αγ − βδ, αδ +…
invalid syntax
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Why is matrix multiplication associative?

When I wonder why the product of elements of symmetric group of order 3 are associative, I just say they are isomorphic to permutation matrices and they just share that feature. Don't take me light....
Allan Baker
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Commutative operation - no simplifications

Suppose we have a set with an associative operation, but we don't know the set is a group. Therefore no simplifications are possible. We want to prove that if there exists $n \geq 2$ such that for any two elements $x,y$ in the set we have $(xy)^n =…
Beni Bogosel
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What is a typical and clarifying example of a set that satisfies associativity?

I understand what associativity is, and of course simple structures from elementary and high school mathematics like the natural numbers over addition are associative. However, I think it would be clarifying to have an example of a mathematical…
user56834
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Is $\mathbb{Z}/n\mathbb{Z}$ a group under multiplication when $n$ is prime?

I have to prove that $\mathbb{Z}/n\mathbb{Z}$ is not a group under multiplication for all n>1. I would argue, however, that when $n$ is prime, $\mathbb{Z}/n\mathbb{Z}$ is a group under multiplication. Multiplication is associative, every element in…
Obliv
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Are there interesting examples of distinct operations that associate?

I was just reading a text on first-order modal logic and found it very interesting to think of the distinction between rigid and non-rigid designators as a failure of associativity between the modal operator of, for instance…
Addem
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tax and discount problem

Problem : In the total purchase amount $z, x\%$ is tax and $y\%$ is discount. Even if the tax is applied first and then discount or if discount is applied and then tax, the final amount is always same. How to prove this mathematically ?
Karthik
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Associative functions of real numbers with 1 and 0

What are the binary functions $F$ of the real numbers, possibly taking an open subset or including infinity, that have an identity element and a zero element and are associative? I know that $F:[-\infty,\infty)^2\to\mathbb R, F(x,y)=x+y+m$ works…
1Rock
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Consequence of the associative law of multiplication

Everyone knows that a * (b * c) = (a * b) * c But how to prove the same rule for more factors? I know how to do this for 4, 5 factors separately. But how to prove right away that any arrangement of brackets in an expression from any number of…
VladimirVlasov
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