# 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.

526 questions
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### Abstract Algebra - Binary Operation [on hold]

Determine whether the definition of * does give a binary operation on the set. If it is a binary operation, determine whether if is commutative and associative. On Z, define * by a*b=a^b
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### How to prove formally that division is not commutative?

I am studying quantitative reasoning and I am wondering how to prove formally that division is not commutative. Everyone knows that 7/3 is not the same as 3/7, but is there a formal way to ...
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### What is the significance of the precedence of a semiring/ ring operations?

In general, a semiring/ring is equipped with two distinct binary operations, namely addition $(+)$ and multiplication $(×)$ , where in most of the cases, multiplication distributes over addition and ...
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### Does this algebraic structure belong to a subclass of neofield (or other structure)?

This article indicates that fields may be subclasses of neofields to the extent that they are neofields with associative addition. The author uses the neofield structure exclusively with finite sets, ...
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### Should we distinguish the minus sign from the negative sign?

In the set $\mathbb{C}$ of complex numbers, the minus sign "-" may be used for following: As a unary operator $-_u$, given a complex number $a$, $-_ua$ is the unique number (called the negative of a) ...
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### Known Algebraic Structure(s) by Operation and Element Characteristics

Is there a known algebraic structure with two binary operations defined such that one of the operations behaves like multiplication (identity, distributive, commutative, and associative) and the other ...
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### Does invertability and closure imply identity?

Sorry if this is a basic question or I'm overthinking it, but if an algebraic structure has inverse elements (or at least for a member $a$), that means $a^{-1}a=e$, and if there's closure then e is an ...
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### A question from 1989 leningrad mathematical olympiad

Prove that we cannot define an binary operation $*$ on the set of integers Z satisfy all of the three properties below simultaneously: For any $A∈Z,B∈Z,C∈Z:$ 1.$A*B=-(B*A)$ 2.$(A*B)*C=A*(B*C)$ (...
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### Is the division symbol $\div$ acceptable based on international standards? [closed]

The division symbol $\div$ is found in almost all calculators; however, I seldom see it in any formal writing. It seems people almost exclusively prefer $\frac{a}{b}$, $a/b$ or $ab^{-1}$ to $a\div b$. ...
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### Question about order of operations convention and alternative

I'm (trying) to make my own code parser that evaluates expressions. I've done some reading about this topic and seen the various mnemonics that school children are taught, such as: P(arentheses) E(...
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### a conjecture on the binary operation of multiplication [duplicate]

Give a conjecture describing the values of $n$ for which all of the nonzero elements of $Z_n = {0, 1, 2, . . . , n − 1}$ have multiplicative inverses. I am guessing the point of not having $0$ ...
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### A well-defined binary operation on a class of functions (Eudoxus magnitudes) from $\mathbb N$ to $\mathbb N$?

Update: The answer shows that some tweaking is necessary to get this to work. The problem are those $f$ where there exist an $N$ such that $f(n)$ is always odd for $n \ge N$. But this can be remedied ...
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### Is there ANY context in which f(x,x) is noncommutative?

Stupid question, but one occasionally reads such things as "the operation $\ast$ is noncommutative for all $x,y$ such that $x\neq y$" or "$x\ast y$ is commutative iff $x=y$". These statements bother ...
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### Why should a non-commutative operation even be called “multiplication”?

As per my knowledge and what was taught in school, $a\times b$ is $a$ times $b$ or $b$ times $a$ Obviously this is commutative as $a$ times $b$ and $b$ times $a$ are same thing. On the other ...
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### Bounds of defining operations for equivalence classes

When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific ...
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### Is there a set in which division of 0 by 0 is defined?

The reason I ask this is that I've discovered that, even though they don't satisfy all field axioms, there are sets called projectively extended real number line and Riemann sphere, which are ℝ∪{∞} ...
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### Function/Operation Definitions

If I want to define a function/operation for equivalence classes, is it permissible to stipulate contingencies based upon element characteristics or does dependency on the choice of element ...
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### An $m$-ary function that represents all $n$-ary functions

Motivation It is well-known that any binary operator $*$ on the boolean ring $\{0,1\}$ can be represented using only one of the $\operatorname{NAND}$ and $\operatorname{NOR}$ operators. For example,...
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### Definition of “fixed point” for binary operator

What is a term for a value $x \in X$ that for binary operator $f\colon X\times X \to X$ maps always to itself, no matter what is the other value $$\forall y \in X \quad f(x, y) = f(y, x) = x$$ Is ...
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### Help Me Understand the Generalisations of Commutativity and Associativity for Unions

Introduction I can prove the generalisation of commutativity and associativity of unions, but I do not understand how they are generalisations of commutativity and associativity. I think it would ...
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### Inverse operation of xor

If x = a xor b, given the values of x and a can we find b? In other words, which function can be applied on both sides in the equation to get the value of b?
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### Floating Point to Decimal Conversion

If a floating-point number is stored in one byte such that the first bit is the sign, the next three bit represent the exponent in excess-4 notation, and the last four bits represent the mantissa, ...
### Is $f(g,h) = g^2$ a binary operation on a group $G$? [closed]
If $(G,\cdot)$ is a group, then $f : G \times G\to G$, defined by $f(g, h) = g^2$, is a binary operation on $G$. Is it true or false and why?