Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [binary-operations]

A binary operation on a set $X$ is a map $\ast : X \times X \to X$. Usually, we denote $\ast(x, y)$ by $x\ast y$. For questions about operations in binary arithmetic (base 2), use the tag (binary) instead.

Note: Whether or not a given operation is a binary operation can depend on the set. For example, subtraction is not a binary operation on $\mathbb{N}$ but it is on $\mathbb{Z}$.

There are many objects in abstract algebra which require binary operations as part of their definition. These include: magma, semigroup, monoid, quasigroup, ring, and field.

An $n$-ary operation on $X$ which is a map $\ast : X^n \to X$. A binary operation is the special case $n = 2$.

875 questions
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Find the number of solutions for $\underbrace{x*x*\ldots*x}_{x\text{ 10 times}}=\frac{1}{10}$ with a given binary operation.

On the interval $(-1, 1)$, consider the binary operation $$x*y=\dfrac{2xy+3(x+y)+2}{3xy+2(x+y)+3}$$ with $x, y \in (-1, 1)$. I have to find the number of solutions for the equation: $$\underbrace{x*x*\ldots*x}_{x\text{ 10…
user592938
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What is Binary Operation -- is division a binary operation?

I was reading the definition of a binary operation of here. The thing I don't understand is how is division a binary operation? If you consider division with pairs in $\mathbb{N}_{>0}\times\mathbb{N}_{>0}$, you do not neccesary get an elenment in…
Adeeb
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Algebraic Structure: Are Set Operations Considered Binary Operations?

I'm currently trying to understand the "hierarchy" of sets / algebraic structures, e.g. things like groups, rings, fields, modules, algebra, vector spaces which I mostly understand, but especially the more technical things like boolean algebras…
EthanAlvaree
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For any function $f$, could one always define an operator $\circ$ so that $f(a\circ b)=f(a)\circ f(b)$?

Given any function $f$, I'm interested in knowing whether one could always define an operator $\circ$ (that is, a two place function) such that $$f(a\circ b)=f(a)\circ f(b)$$. Now, if there exists some element $c$ in the domain of $f$ such that…
Sam
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Binary operation defined as $a*b=a+b-ab$

Binary operation '*' defined as $a*b=a+b-ab$ on a set $A=R\setminus\left\{0,1\right\}$ where $R$ is Set of Reals Is this binary operation closed under the above operation on above set $A$ I thought it is not closed since if $a=b=2$ then $$2*2=0$$…
Umesh shankar
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How many adders needed in order to implement $y=x^2$?

We are given a natural number $x\in\mathbb{N}^+$, and $x-1$ adders. An adder in our case would be a component with two inputs $x_1,x_2$ and a single output $y$ such that $y=x_1+x_2$. Our goal is to design a system that has $x$ as its single input,…
Amit Zach
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How do we deal with units when using the modulo operation?

I'm wondering how I should deal with units when I do a modulo operation. What is considered legal and what is not. When I have two numbers that have units such as 13cm and 3cm, I can multiply them: $$13cm \times 3cm = 39cm^2 $$ Similarly, I can do…
Alexis Wilke
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Can all operations on multiple elements in mathematics be reduced to binary operations?

To take a very simple example: imagine the sum 1 + 2 + 3 + 4. You can do this one step at a time: 1 + 2 = 3, then 3 + 3 = 6, then 6 + 4 = 10. It does not matter how long it takes you to move from one step to the next. The answer will always be the…
James Newton
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Associative Numbers

Yesterday my friend wrote a number on a paper. He then added the number of ones in the binary representation of the number to that number and formed a new number. He kept doing the process infinitely. So he got a chain of numbers say A->B->C->D…
Kme
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No Identity Element

For $x, y ∈$ $\mathbb{R}$, let $x△y = 2(x + y)$. Then $△$ is a binary operation on $\mathbb{R}$. Show that there is no identity element for $△$ on $\mathbb{R}$. I have tried $x△e = e△x=x$ I don't know what else to do.
mku
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Need help confirming which algebraic structure this is

Let's define a binary operation $*$ on $\mathbb{R}$ such as $$ a * b = e^{a+b} $$ and investigate which algebraic structure this is. Well first of all we notice that the operation is closed under $\mathbb{R}$ since if $a,b \in \mathbb{R}$ and $e^x$…
Markus Punnar
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Binary Operations (Commutative and Associative).

Let $*$ be a binary operation on $\mathbb R$ given by $x*y = (x^{1/3}+ y^{1/3})^3$. Determine if it is commutative and associative. I know how to prove it is commutative. \begin{align*}x * y &= (x^{1/3} + y^{1/3})^3\\ &= (y^{1/3} +…
Tiffany
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Define a binary operation on the set of even integers which is different from addition,substraction and multiplication

Question: Define a binary operation on the set of even integers which is different from addition,substraction and multiplication. My attempt: $a \circ b = a + b - ab$ where $a\in G$ and $b\in G$ ($G$ is the set of all even numbers) Is my answer…
Shampa Das
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closed non associative binary operation

I'm Trying to show that a binary operation does not have to be associative in order to maintain closure, identity element and inverse element - on a 5 elements set. Any ideas for a set + binary operation that shows that? I can't find a way to lose…
Hilla
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Simple example of idempotent but not commutative nor associative binary operator?

Is there a simple example of a binary operation that is idempotent, but not commutative nor associative?
Filip Haglund
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