Questions tagged [associativity]
This is the property shared by many binary operations including group operations. For a binary operation $\cdot$, associativity holds if $(x\cdot y)\cdot z = x \cdot(y\cdot z)$.
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How do you prove associativity for this operation?
Let $S$ be a set with a binary operation $/:S\times S \rightarrow S$ where $(a,b) \mapsto a/b$ such that:
There exists an element $1 \in S$, such that $a/b = 1$ if and only if $a=b$.
For any ...
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priority of operations in function composition is backwards
I feel like the priority of function composition is backwards, and I would like to have a deep understanding of the phenomenon.
I do understand that function composition reads right to left:
$$(f\circ ...
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Practical example of differences between associativity and alternativity (and the in-between Bol loop)? [closed]
Associativity is:
$$(a * b) * c = a * (b * c)$$
Alternativity is:
$$a * (a * b) = (a * a) * b$$
$$(a * b) * b = a * (b * b)$$
Bol loop is:
$${\displaystyle a(b(ac))=(a(ba))c}$$
$${\displaystyle ((ca)b)...
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Proof for associativity of a modified "matrix multiplication"
I'm trying to prove the associativity of a modified form of matrix multiplication (which is defined below) and I found the following proof which I'm confused about:
For matrices $W_i, W_j, W_p$, to ...
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Why is 1+2+3 = (1+2)+3 [duplicate]
Not sure if this is a stupid question, apologize if it is.
I am curious why we can add the first 2 numbers, then add the third one when doing addition of 3 numbers.
There is a similar (IMHO) question, ...
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A pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ such that $f$ and $g$ are idempotent, commute with each other and $f \times g$ is bijective
The question
The question is:
Does there exist a pair of mappings $f, g: \mathbb{N} \to \mathbb{N}$ satisfying the following properties?
$f$ and $g$ are idempotent, meaning that $\forall n \in \...
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why is this associative?
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already 😅
The fourth exercise asks me to determine whether the following operation is compliant with the associative ...
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$(ab)c + a(bc) = 2 b (ac) \implies^? x(yz) = (xy)z$? [closed]
Consider some unital commutative algebra $A$ such that for all its elements we have
$$(ab)c + a(bc) = 2 b (ac) $$
Does this imply the algebra is associative ?
or in symbols :
$$(ab)c + a(bc) = 2 b (ac)...
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Proving Associativity of the Sum in a Space of Infinite Sequences with Non-Zero Initial Element
Consider the set $V$ consisting of all infinite sequences $a = (a_0, a_1, \ldots)$ where each $a_i \in \mathbb{R}$ and $a_0 \neq 0$. How can we demonstrate that the operation $(a + b) + c = a + (b + c)...
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Do we ever reason about a non-associative algebra without embedding it in an associative algebra?
This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math.
...
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Can any binary operator be turned, through an associative operator, into another associative operator?
Motivation: let $\circ:X^2\to X$ be some binary operator, and let $+:X^2\to X$ be some commutative operator. Then
$$\star:X^2\to X:(x,y)\mapsto (x\circ y)+(y\circ x)$$
is commutative.
I was wondering ...
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When can a partial associative operation be extended?
Let $X$ be a set with a partial operation $\cdot$ which is associative in the sense that if $x, y, z \in X$ and $x \cdot y$ and $y \cdot z$ are both defined, then $(x \cdot y) \cdot z$ and $x \cdot (y ...
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Showing matrix multiplication is associative via linear mappings.
Exercise.
Prove that matrix multiplication is associative. In other words, suppose $A, B$, and $C$ are matrices whose sizes are such that $(AB)C$ makes sense. Explain why $A(BC)$ makes sense and prove ...
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Is this 3D algebra $T$ power-associative?
Before reading this question it is essential that you understand power associativity
https://en.wikipedia.org/wiki/Power_associativity
In particular a commutative algebra does not necc imply a power-...
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Abstract formulation of associativity
Say we are given a binary operation $f$ on a set $X$, that is,
$$
f : X \times X \to X.
$$
Denote by $\text{Id}$ the identity map on $X$.
We say that $f$ is associative if, for all $x, y, z \in X$, we ...
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