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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Possible outcomes of sums
Let $\mathrm{S}$ be a set of distinct 20 integers. A set $\mathrm{T}_{\mathrm{A}}$ is defined as $\mathrm{T}_{\mathrm{A}}:=\left\{\mathrm{s}_1+\mathrm{s}_2+\mathrm{s}_3 \mid \mathrm{s}_1, \mathrm{~s}...
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Uniqueness of Binary Operations
I was doing an AMC question that used a binary operation, ?, that was defined as such:
a?(b?c) = (a?b)*c , for all real, non-zero a,b,c, where * is normal multiplication. In addition, (a?a)=1 for all ...
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Can a vector with unordered components exist?
It seems like in order for a vector addition to be commutative, it needs to be defined in a "regular" manner, i.e. by adding matching vector components (because then the commutativity of ...
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Is this definition of commutativity correct?
Everywhere I see commutativity defined somewhat like this:
A binary operator $*$ is commutative in $S$, if for any $x, y \in S$, the following property holds: $x*y=y*x$.
That definition is, of ...
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Expected value of $3 \circ 3 \circ 3 \circ \dotsc \circ 3$, with $\circ \in \{+,-,\times,\div\}$
You are given an expression with $n$ 3's and $n-1$ $\circ$'s and you wish to evaluate
the expected value of $3 \circ 3 \circ 3 \circ \dotsc \circ 3$, with $\circ \in \{+,-,\times,\div\}$.
What I did ...
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Determining the binary operation from a Cayley Table
I have tried a lot of things with this Cayley Table (several Julia/Python scripts which iterate over various functions, symbolic regression, semi-manually trying various permutation groups, octonions, ...
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error in textbook exercise regarding binary operations?
The following exercise of from Guide to Abstract Algebra by Carol Whitehead, 1st Edition 1988.
Let $\bullet $ denote a binary operation on a non-empty set $S$. Suppose that $\bullet $
admits a left ...
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Division of binary numbers, confusing
I am trying my best to divide the following:
Perform the following computations in binary arithmetic (Show how you perform
the computations):
My attempt:
I watched:
https://www.youtube.com/watch?v=...
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A function on binary vectors
I am looking for a function $f:\{0,1\}^d \to \{0,1\}^{d'}$, where $d'<5d$, such that whenever $x,y,z$ are three distinct $d$-dimensional binary vectors, and for all $i\in\{1,\dots, d\}$ it holds ...
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Doubt on complexity of multiplying two binary numbers.
In one of my lectures, it was stated that when we multiply two binary numbers (say $n$ and $m$) such that $n$ has $k$ bits and $m$ has $l$ bits we have a maximum of $kl$ bit operations.
I don't ...
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If a binary number is greater or equal than another one, does it have the same or more number of digits?
This question came up to my mind while dealing with complexity of binary operations, speciffically with multiplication.
The question. Let's say we have two binary numbers $n$ and $m$, such that $n \...
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Whether binary operations are closed or not?
I recently studying Gallian Abstract Algebra 8th edition by Pearson Book. And found an exercise of chapter "group" that asks to find binary operations which are closed.
But by the definition ...
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use of binary operation to represent hadamard transform
In very frank terms, meaning basic binary operations and well-defined definitions, can someone explain to me what this operation is supposed to mean and what the notation is supposed to represent? I ...
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Showing a non-unimodular group operation is not associative
There is a binary operation defined by
$$(f*g)(x)= \int_G f(xy^{-1})g(y)dy$$
where G is not a unimodular group.
Show is operation is not associative.
Workings so far:
$$((f*g)*h)(x)=\int (f*g)(xy^{-1}...
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Is there a name for the number of ones in a binary vector?
I'm creating two types of binary vector. The vectors have $M$ components and either $N$ ones or $(M-N)$ ones.
If you were to take all permutations of these vector types there would be a symmetry in ...
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A combinatorial problem with coins
I am stuck at a mixture of a combinatorial and maximization problem and don't know how to proceed. Hopefully someone has an idea that can bring me further.
Imagine that we have a sequence of $n$ coins....
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Does there exist a group $G$ with a proper subgroup $K$ in which for all $a, b \in G - K$ such that $ab \neq e$, we have $ab \in G - K$? [closed]
I am currently studying group theory and I asked myself various questions, though I was able to solve almost all of them, I could not answer the following one:
Does there exist a group $G$ with a ...
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Binary math for with decimal numbers [closed]
How do you make the math operation to represent float numbers?
if example I want to represent 0.2 in binary.
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Map obtained by combining two binary operations
Consider a set $A$ as well as two binary operations $*_1$ and $*_2$ defined on $A$. Is there a name to describe maps which are defined fully in terms of $*_1$ and $*_2$? For instance $f:A^4\to A$ ...
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No Identity Element [closed]
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 ...
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Question about what I am allowed to do in a finite set which has the closure property under a certain operation [closed]
I have a question regarding what I am allowed to do whenever I have a set of elements from a ring that has the closure property. Suppose we have a ring $(A, +, \cdot)$ and we consider the finite set $...
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Comparative Function for reals
In Computer Science, it is common to use a comparative function like $cmp(a, b)$ which returns either -1, 0, or 1 based upon the relative values of a and b. Commonly a result of -1 (or just less than ...
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In group theory, are the operations left to right or right to left?
I got points taken off on homework because for the dihedral group $D_3$, I needed to find the left coset of a subgroup $H$. So for "flip $\cdot$ rotation" I did the flip first and then the ...
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About 2 operators such that $X*(Y+Z) = (X*Y)+(X*Z)$ , $W+(U+V)\neq (W+U)+V$
This question is a follow-up and analogue of a previous very similar question.
I have to seperate questions since it is a bit of a rule ; 1 question at a time , hence this seperate one.
Looking at the ...
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About 2 operators and $A*(B+C) =(A+B)*(A+C)$
Consider $2$ binary operators defined for a finite set with $n$ elements.
Operator $*$ behaves like a commutative latin quandle :
$$x*x = x$$
$$a*b=b*a$$
$$a*(b*c)=(a*b)*(a*c)$$
And forms a latin ...
- 14.1k
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Obscure properties of binary operations
The commutative, associative, and distributive properties of binary operations are very well-known. I am interested in more obscure properties of binary operations, which have nonetheless been studied ...
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Why is Closure a necessary axiom in Group Theory? [closed]
How can we explain the logic behind the closure axiom in Group Theory to a non-math major? What is the reasoning for why it is necessary? If the result of compounding two elements falls outside the ...
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Product between two non-square permutation matrices
Given a $n \times m$ permutation matrix $X$ (with $0 < n <= m$), I would like to show that $X'X$ must be diagonal and $XX'=I$. How can I prove that?
Note that a permutation matrix is a square (0,...
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Converse to a proposition regarding associative and switchable binary operations.
I define the switchability property of binary operations as follows: An ordered pair $(+,*)$ of binary operations on a set $S$ is said to satisfy switchability if for all $x,y,z$ in $S$, $(x+y)*z=x+(y*...
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What makes multiplication a basic operation on natural numbers but exponentiation is not?
Dear StackExchange Math Community:
It has puzzled me for some time why multiplication is considered a basic arithmetic operation on natural numbers, but exponentiation is just viewed as a shorthand of ...
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A question about switchable binary operations
Suppose we are given that $ * $ is an associative binary operation on a set $S$. Suppose further that $\times$ is a binary operation on $S$ such that this property holds, which I call switchability: ...
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Is this a sufficient condition for a binary operation on functions to be a pointwise operation?
Let $S$ be an arbitrary set, and let $F$ be the set of all functions from $S$ into $S$. A binary operation $*$ from $F \times F \rightarrow F$ is said to be a pointwise binary operation if there is a ...
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Find all $scattered$ numbers less than $2^n$ whose in its binary expansion there are never two 1's immediately next to each other
Any positive integer can be written in binary (also called base $2$ ). For example, $37$ is $100101$ in binary ( because $37=2^5+2^2+2^0)$, and $45$ is $101101$ in binary. Let's say that a positive ...
user1086101
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Bit reversal problem
I'm trying to figure why divide and conquer technique has this property:
for example, we count 0 to 7, then we divide them into 2 parts: [0, 2, 4, 6] and [1, 3, 5, 7]. Then we divide each of the 2 ...
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Can the "square and multiply" and "double and add" algorithms be generalized to just about any power-associative algebra?
Both the "square and multiply" and the "double and add" algorithms are optimizations for what can be naively computed using a series of binary operations, on some initial value $p$....
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How to take absolute value in docplex without calling the built-in function?
I have the Constraint:
$$
| \sum x_{i} - n/2 | \leq 0
$$
where each $x_{i}$ is a binary variable, and $n$ is the total number of variables.
in docplex, I encode this as:
...
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What is the most efficient way to "encode" a finite field into a structure that has only one binary operation?
I know that a finite field can be easily encoded into and decoded from a finite Paige loop, which can be defined as a Moufang loop that is simple and isn't a group. However, the finite Paige loop of a ...
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Find an operation so that it makes $D^{2}=\{z\in \mathbb{C}\mid |z|<1\}$ a group.
Let $D^{2}=\{z\in \mathbb{C}\mid |z|<1\}$. Find operation $\oplus: D^{2}\times D^{2}: \rightarrow D^{2}$ so that it makes $D^{2}$ a group.
I have already defined some operations but these don't ...
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Has this strange real-valued binary function been researched?
Let $X \neq \emptyset$ be an arbitrary set. Let
$$ d : X \times X \to \mathbb{R} $$
be a real-valued function with the following properties:
(1) $\forall x, \in X: d(x,x) = 0,$
(2) $\forall x,y \in X:...
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Function application
We can create a function (let me call it $¬$) that maps a function and it's input to the value of $f$ at $x$, look at this previous Previous question of mine I discuss the idea of this, my question is ...
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Question about the definition of a homomorphism
In Fraleigh's abstract algebra book, he gives definitions on how structure carries over between two isomorphic binary structures. The first definition is given relatively early in the book (Section 3) ...
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Operation 'Referencing' In Abstract Algebra
I recently started an Abstract Algebra course so I may not know all of the 'necessary' vocabulary to fully express my question, but here is my attempt. First I need to provide some background thinking....
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Question on binary relationship.
Condition A:
Given x, y in X such that $yRx$ then it follows that
$\lambda y +(1-\lambda)xRx$ for all $0< \lambda<1$
Condition B:
Given x, y in X such that $yPx$ then it follows that
$\lambda y +...
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4
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Ambiguity of addition
In the case of addition, we are taught from early to just put a $+$ between the addends, 'this is $2+2$ this means $4$' and hence we define the value of $2+2$, however, I've seen 'addition' defined as ...
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Binary relation vs Binary operator?
I'm confused about the difference between a binary relation and binary operator, a function is a specific example of a binary relation, a binary relation $R$ could be a function, and a binary operator ...
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Question about one of the definitions of an isomorphism between two binary algebraic strutures
From Fraleigh's Introduction the Linear Algebra, there is one particular characterization of an isomorphism that I am confused on.
Have $\langle S, *\rangle$ and $\langle\dot S, \dot *\rangle $ be ...
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Why addition/multiplication are not considered a mathematical functions?
Why we always consider exponentiation/logarithm to be a functions of two variables, but the same terminology never applies to addition/multiplication. Is it just because it's never useful or am I ...
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45
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Binary Function on an unordered pair?
We define binary functions such as addition over the reals can be defined like this in the various notations,
$+(a,b)=a+b=b+a=+(b,a)$ in this case there is a separate mapping as for the tuples $(a,b)$ ...
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1
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Mathematical definition of the function to extract a value in an integer by bit selection
I need the definition of a function $s(v,p,w) = i$ where $v$ is an integer that can be expressed as binary or decimal, $p$ is the position of a bit in $v$, and $w$ is an amount of bit from 1 to $n$. ...
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non-commutative algebraic structure with 16 elements, need help categorizing it and finding a representation
We have an abstract algebraic structure with the following multiplication table, has anyone seen this structure before and can anyone give it a proper name and a simple (possibly matrix) ...
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