# 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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### 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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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}... 6 votes 4 answers 2k views ### 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 ... 1 vote 0 answers 72 views ### 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.... 0 votes 1 answer 56 views ### 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 ... -3 votes 1 answer 39 views ### 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. 1 vote 0 answers 15 views ### 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 ... 2 votes 2 answers 57 views ### 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 ... -2 votes 1 answer 68 views ### 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 ... 0 votes 0 answers 27 views ### 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 ... 1 vote 1 answer 91 views ### 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 ... 1 vote 0 answers 58 views ### 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 ... 3 votes 1 answer 74 views ### 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 = xa*b=b*aa*(b*c)=(a*b)*(a*c)$$And forms a latin ... 0 votes 0 answers 35 views ### 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 ... 1 vote 4 answers 185 views ### 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 ... 0 votes 0 answers 31 views ### 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,... 0 votes 1 answer 33 views ### 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*... 1 vote 1 answer 105 views ### 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 ... 0 votes 0 answers 13 views ### 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: ... 1 vote 1 answer 26 views ### 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 ... 2 votes 1 answer 1k views ### 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 ... 3 votes 1 answer 79 views ### 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 ... 2 votes 0 answers 123 views ### 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.... -1 votes 1 answer 62 views ### How to take absolute value in docplex without calling the built-in function? I have the Constraint:$$ | \sum x_{i} - n/2 | \leq 0 $$where each x_{i} is a binary variable, and n is the total number of variables. in docplex, I encode this as: ... 4 votes 2 answers 107 views ### What is the most efficient way to "encode" a finite field into a structure that has only one binary operation? I know that a finite field can be easily encoded into and decoded from a finite Paige loop, which can be defined as a Moufang loop that is simple and isn't a group. However, the finite Paige loop of a ... 0 votes 1 answer 37 views ### Find an operation so that it makes D^{2}=\{z\in \mathbb{C}\mid |z|<1\} a group. Let D^{2}=\{z\in \mathbb{C}\mid |z|<1\}. Find operation \oplus: D^{2}\times D^{2}: \rightarrow D^{2} so that it makes D^{2} a group. I have already defined some operations but these don't ... 1 vote 1 answer 85 views ### Has this strange real-valued binary function been researched? Let X \neq \emptyset be an arbitrary set. Let$$ 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) $\forall x,y \in X:... 0 votes 1 answer 51 views ### Function application We can create a function (let me call it$¬$) that maps a function and it's input to the value of$f$at$x$, look at this previous Previous question of mine I discuss the idea of this, my question is ... 4 votes 2 answers 681 views ### Question about the definition of a homomorphism In Fraleigh's abstract algebra book, he gives definitions on how structure carries over between two isomorphic binary structures. The first definition is given relatively early in the book (Section 3) ... 2 votes 3 answers 133 views ### Operation 'Referencing' In Abstract Algebra I recently started an Abstract Algebra course so I may not know all of the 'necessary' vocabulary to fully express my question, but here is my attempt. First I need to provide some background thinking.... 0 votes 1 answer 84 views ### Question on binary relationship. Condition A: Given x, y in X such that$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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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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### 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 $(a,b)$ ...
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 $n$. ...