# 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.

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### Counting binary operations on a group [closed]

For a finite set S with n elements, how many binary operations can be there which makes S a group?
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### Can every group be embedded in a non-associative group? [closed]

A non-empty set S that satisfies all group axioms except associativity is called a non-associative group. If $G$ is any group then is it possible to embed $G$ in some non-associative group?
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### Number of steps needed to show that a binary operation is associative on n operand

I was trying to prove a binary operation is associative on a given number of the operand. I did it, but then I checked the textbook. I took 5 lines, but the book proved it in 6 lines using a precise ...
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### Are Linear Complexes and Simple Graphs the same thing?

I refer to the background of the notion of $Linear~complex$ by A. A. Zykov in the article entitled "General properties of Linear Complexes" A $Linear~complex$ (or simplex complex) is a ...
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### Associativity of binary operations on a two-element underlying set (is there a pattern?)

The overall problem is to establish, which binary operations on a two-element set $A=\{a, b\}$ are commutative and associative. There are 16 of them altogether, obviously, analogous to operations on ...
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### For $G_1$, $G_2$, $G_3$ simple undirected graphs on the same vertices with disjoint edge sets, if $G_1\cup(G_2\cap G_3)=G_2$, then $G_1=G_2$

If $G_1, G_2$ and $G_3$ are simple undirected graphs on the same set of vertices with disjoint edge sets. If we have a graph equation $$G_1\cup(G_2\cap G_3)=G_2$$ Then we have to show that $G_1=G_2,$ ...
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### How can I prove that an invented operation works with negative numbers? [duplicate]

Just out of curiosity I tried making a new operation to see its properties. It is like a variation of addition that works like this: a ¨ b = a + b - 1 For example, I found that it has the commutative ...
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### Does the set intersection operation has unit

I'm trying to get better understanding of binary operations, and I came across this problem: namely on one online discussions I saw that set intersection as binary operation doesn't have a unit, ...
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### Are all commutative, associative binary operations isomorphic to addition?

Addition and multiplication are the two classic commutative, associative binary operations on the reals. They satisfy a striking property: they are equivalent up to unary operations. By taking a ...
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### Operations on functions with different co-domains

How are operations on functions with different co-domains defined? Let f: D1 -> C1 and g: D2 -> C2, and let (f + g): D -> C What are sets D and C in terms of D1, D2, C1, C2? Furthermore, is ...
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### Are these variations of complex multiplication studied topics?

Complex multiplication is very well understood geometrically and algebraically, but I wonder what about the following operators -angles assumed to be randians $[0,2\pi)$: Complex multiplication(...
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### finding a four elements group and binary operation upon it that has closure, associativity, an identity element and an inverse to each element.

the question is in the heading. I tried using the group {0,1,2,3} and the binary operation $*$ defined as $a*b=|a-b|$ which I proved to have closure, an identity and an inverse, but am unable to prove ...
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### Confusion about binary expansion of number in $(0,1)$.

Unfamiliar with discrete mathematics, I wondered in the following two answers Fast way to find period-n points of a tent map? Techniques for finding period points , why the authors wrote: 1. "...
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### Let $(A,+, \cdot)$ be a ring with the given tables, determine which is the zero and inverse of each element

Let $(A,+, \cdot)$ be a ring with the given tables for operations $+$ and $\cdot$ \begin{array}{c|cccc} + & s & t & x & y \\ \hline s & y & x & s & t \\ t & x &...
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### Given a set A in which all elements are of the form: $x+y \sqrt3$; $x,y \in \mathbb{Q}$. What structure can you define…

Given a set $A$ in which all elements are of the form: $x+y \sqrt3$, $x,y \in \mathbb{Q}$. What algebraic structure can you define with operations of addition and multiplication? I am stuck figuring ...
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### Does distributivity implies commutativity of one operation

Suppose there is a set $S$, equipped with two binary operations, $*$ and $@$, such that S is closed and associative under both the operations. There exist inverses and identity with respect to both ...
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### Why 1 isn’t $a*b=max(a,b)$ identity element?

I had an exam this morning and the first question was in a test form ,the question goes: Imagine there is a binary operation $a*b=max(a,b)$ on N . Which option is correct ? a)$(N,*)$ is not abelian. ...
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### Find path through binary tree to get to desired node #

Say we have a binary tree. If know how many nodes X there are in the tree, how can we navigate from the root node to the node with value X without any backtracking? I am doing binary arithmetic and ...
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### Given, (F,+,•) and (F,+,×) are fields; then • and × are same binary operations

(While reading about classification of simple groups, I was wondering if there could be something like simple fields or something... And this thought came in mind) When we talk about field in abstract ...
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### Addition vs multiplication as binary operators

It is known that arithmetic operations of addition and multiplication are binary operations that take in two inputs and give out a single output. However, consider the following scenario: Suppose I ...
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### Arithmetic operation

Given $$9 ∆ 4=20,$$ $$7 ∆ 3=12,$$ $$10 ∆ 8=16,$$ find $m ∆ n$. I have tried so many ways to get this but I may be missing something.
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### Is there a deeper reason why exponentiation is not associative?

Addition can be thought of as repeated counting; multiplication can be thought of as repeated addition; and exponentiation can be thought of as repeated multiplication. And yet, while the first three ...
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### For which type of input does this operation repeat the outcome eventually and run forever?

This problem is a part of another (programming) problem on which I am working, the start of which requires us to identify a certain pattern, I believe. We are given a pair of positive integers x and y....
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### Binary operation or Binary relation? [closed]

$Y$ corresponds to set of all the binary relations over set $X$. If $R$ is the composition of binary relation on set $X$, is $R$ a binary operation or a binary relation?
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### Bijectivity of a binary function

How to prove that any binary function (Boolean function) is bijective in $x$ if and only if the function is linear in $x$. For example, $f(x_{1}, x_{2}, ..., x_{n}) = x_{1} + x_{2}*x_{3}*...*x_{n}$ is ...
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### How does all 1 exponent and all 0 fraction represent infinity?

I am studying IEEE 754 FLOATING POINT STANDARD. Standard says The number is infinity when: e (Biased Exponent) = 255 f = 0 I am unable to understand this because if fraction (f) = 0 and exponent = ...
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### Is it possible to define some form of relationship between two operations?

I am thinking we have transformations and all forms of relationships on sets. But thinking for an abstract algebraic theoretical session, I pondered on whether its possible to transform (or just ...
Consider a binary operation $*$ acting from a set $X$ to itself. It's useful and standard to work with operations which are associative, such that $(a*b)*c = a*(b*c)$. What about operations which are ...