A binary operation on a set $X$ is a map $\ast : X \times X \to X$. Usually, we denote $\ast(x, y)$ by $x\ast y$.

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Proving an idempotent binary operation where $(x\ast y)\ast z=(y\ast z)\ast x$ is commutative

Let $S$ be a set and $\ast$ be a binary operation on $S$ satisfying 1) $x\ast x=x$ for all $x\in S$, 2) $(x\ast y)\ast z=(y\ast z)\ast x$ for all $x,y,z \in S$. Show that $x\ast y=y\ast x$. I ...
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Why is Nim solvable with the xor operator?

In the game of Nim, played with two players, if you have $n$ stacks of stones (where you can take any number of stones from a single pile each turn), losing positions are ones where the xor of the ...
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Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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Determine the operation based on the conditions given below

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
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an operation using pit operators [closed]

Hello all i have some numbers and i make some operations in them which give me a number i save in a file so that's the codes ...
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⋇ “Division Times” operator in Unicode (U+22C7)? [duplicate]

I found this maths operator in Unicode: ⋇ It is called "Division Times" (U+22C7). Does it behave like ±? For example: 3 ± 2 means it is an ∈ {1, 5}. So 3 ⋇ 2 means it is an ∈ {1.5, 6}?
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Polynomial ring operations on $\mathbb{Z}$

The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables): Addition: $(a,b) \mapsto a+b \in \mathbb{Z}[a,b]$ Multiplication: $(...
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Induced operation and anticommutativity

Let $\odot$ and $\circledast$ be operations on X and Y. Let $f:X\to Y$ satisfy $f(r_x)=r_y,\ f(x\circledast y)=f(x)\odot f(y),\ x,y\in X$. Prove: $\ x\sim y:\Leftrightarrow f(x\circledast y)=r_y \...
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How to prove that given a binary operator $*$, $a * b = a^2 - ab + b^2$ is associative?

No matter how hard I try I cannot seem to prove that, given the binary operator $*$, the operation $a * b = a^2 - ab + b^2$ is associative. This is what I have tried: $(a * b) * c = a * (b * c) [...
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Does associativity imply closure?

Does associativity of binary operation imply closure under this operation? Sometimes definitions of semigroup, group or vector space omit axiom of closure under corresponding operations and sometimes ...
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29 views

Negative representation of a binary number

I saw online that if you want to convert a binary number to a negative binary number, you add 1.However, I don't understand why you do that.In a forum I saw someone explaining the following: ...
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7answers
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Does associativity imply commutativity?

I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea: $$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ ...
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1answer
17 views

Mathematical Properties of Logical Shift

Many programming languages feature a logical shift operator, such as $(x >> n)$, where the bits of $x$ are shifted $n$ steps to the right (or left, $<<$), and the "vacated" bit positions ...
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24 views

Computation for binary operation

For the binary operation on $x_7$ on set $\{1,2,3,4,5,6\}$, compute $3^{-1} x_7 4 $ Could some tell me what is $x_7$ here so that I may be able to solve this problem?
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33 views

Does the following function define a distance metric?

For real numeric vectors of length $N$, let $a_n \succ b_n$ be one if true and zero if false. The distance between $A$ and $B$ is $$\sum_1^N a_n \succ b_n$$ Note that this is very similar to the ...
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1answer
40 views

Question on the Notation of an Abstract Algebra Question

The following is a question that I came across in a textbook I'm reviewing for self-study. The book is "Introduction to Abstract Algebra", 4th Edition, by W. Keith Nicholson. I have a question both ...
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1answer
48 views

Product of sets as complexes

What does it mean to take the product of two sets of complex numbers as complexes? Reading this paper: "The Determinant of the Sum of Two Normal Matrices with Prescribed Eigenvalues" by N. Bebiano ...
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28 views

Definition of Operation

What are operations in Mathematics? I do not find it formally defined anywhere. What is the difference between operation and function? Earlier I thoght operations are just binary operations. But later ...
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1answer
34 views

Is every such family induced by an associative operation?

Suppose we're given an associative operation $\star : X \times X \rightarrow X$. Then for each $n \in \mathbb{N}_{>0}$, there's a function $f_n : X^n \rightarrow X$ given as follows: $$f_n(x_0,\...
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3answers
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Can $0$ be added to any equation without changing the outcome?

I was thinking about adding $0$ to an equation, e.g.: A very simple one: $$2x + 2 = 10\\ 2x = 8 \\ x = 4 .$$ If you add "$+ 0$" to any side it does not change the outcome. $2x + 2 + 0 = 10 \...
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Is there a name for the operation $f^{-1}(f(x) \oplus f(y))$?

This question is inspired by and/or a generalization of this question about the "reciprocal addition" operation. Consider the following: One is tempted to say multiplication is simply "addition ...
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26 views

Give an example of a set with two binary operations, addition and multiplication, in which we have left distributivity but not right distributivity

Give an example of a set with two binary operations, addition and multiplication, in which the left distributive law holds but the right distributive law does not hold. I.e.: $$a(b+c)=ab+ac\text{, ...
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how to prove $G$ is an abelian group under $*$ (called the real numbers mod 1)

Let $G = \{x \in \mathbb{R}~|~0\leq x < 1\}$ and for $x,y \in G$ let $x*y$ be the fractional part of $x+y$ i.e $x*y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$. ...
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Showing associativity of (x*y) = (xy)/(x+y+1)

In order to show something is associative one must show that $(x*y)*z$ = $x*(y*z)$. I want to show that $x * y = \frac{xy}{x+y+1}$ is associative. This is for self-study (I'm learning algebra over ...
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2answers
70 views

Show that $G$ is a group under $*$

Let $G$ be the set of rational numbers $x$ with $x \neq\frac{-1}{2}.$ For $x, y ∈ G$ define $$x ∗ y = 2xy + x + y.$$ Show that $G$ is a group under $*$. I know how to show that associativity holds ...
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Does this operation have a name?

For a field $F$, define the binary operation $\parallel :(F\mathbb{P}^1 \times F\mathbb{P}^1 \setminus\{(0,0)\}) \to F\mathbb{P}^1$ by $$a \parallel b = \frac{1}{\frac{1}{a} + \frac{1}{b}}.$$ This ...
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1answer
28 views

What is the approach to understand this algorithm?

Given $\{x_1, x_2,\ldots x_n\}$ where $x_i \in \{0, 1\}$ there is a binary equation $\varphi$ that is $x_{t_1}+x_{t_2}+\cdots+ x_{t_m}=0 \mod 2$ where $t_i \in \{1,2,\ldots,n\}$ for $x≥1$, $i=1,2,\...
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Can the state of a system after applying the operation “absolute value” be got back using elementary operations or transformations?

Take the operation or transformation "addition". You can get back the original state of the system by doing the opposite operation, i.e., "subtraction". But, if the operation is "absolute value", you ...
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solving an XOR matrix

I'm working on a somewhat-unique linear algebra problem arising from XORing files together in order to encode them, and then figuring out how to subsequently recreate the original files from the ...
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21 views

Binary Decision Trees

I know the basics to a binary decision tree, but this problem has me a little stumped, and I'm looking for some verification on my ideas. The problem is: "Create a binary decision tree that reflects ...
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1answer
23 views

Anticommutative binary operation not commutative

A binary operation $\circledast$ on a set $X$ is called anticommutative if $\exists r\in X: x\circledast r = x,\;\; x\in X$ and $x\circledast y=r\Leftrightarrow (x\circledast y)\circledast(y\...
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How we can represent $a^b$ in following form

Consider $$a^b= a ^ {101101} $$ As if we split the binary representation of $b$, $$b = 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2+ 0 \cdot 2^1 + 1 \cdot 2^0 $$ Then how are we able to ...
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Why 1/1010 is 0.0001100110011001 [closed]

Can someone please demonstrate why 1/1010 in binary is 0.0001100110011001...? I've tried doing the math and I don't get the same result. Thanks in advance!
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Binary operations being continuous under a topology?

For a set $S$ and some function $f: S \rightarrow S$ and $a\in S$, $f$ is continuous at $a$ under a topology $N$ if for all neighborhoods $N_1(f(a))$ there exists a neighborhood $N_2(a)$ such that $f(...
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Proving that Z with the binary operation is a monoid?

Let $*$ denote the binary operation defined on the set $\Bbb Z$ of integers, where $$x * y = 3xy - 5x - 5y + 10$$ for all integers $x$ and $y$. Prove that $\Bbb Z$, with the binary ...
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1answer
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Binary floating point subtraction

(In binary environment) 0.100011 * 2^6 - 0.111001 * 2^3 = 0.100011 * 2^6 - 0.000111001 * 2^6 = 0.100011000 * 2^6 + 1.111000111 * 2^6 (convert left part into 2's complement) = 10.011011111 * 2^6 ...
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22 views

Characterization of elementary arithmetic operators to explain certain properties in programming languages

In the LISP-like family of programming languages, the four elementary arithmetic operators behave differently: + and * can take ...
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18 views

Induction with associative binary operation

Let * be an associative binary operation on a set 'A' with identity element e. Let 'B' be a subset of 'A' that is closed under *. Let b1, b2, b3, ... bn ∈ B. Prove that b1 * b2 * b3... bn ∈ B. ...
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1answer
21 views

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?
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1answer
31 views

Limits and the Distributive Property?

I am currently relearning calculus after ages of not having much to do with it, and I'm looking at the proofs for the basic limit laws. I was wondering if there is a "simpler"/ more elegant way of ...
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57 views

Number of Distinct Integers in Base-3

Given: An integer $A$ and a natural number $K$. XOR operation in base-3 on bits $x \ and \ y$ is defined as follows, $$ x\newcommand*\xor{\mathbin{\oplus}}y=1,⇔ x=y \\x\newcommand*\xor{\mathbin{\...
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Binary counting problems

Recall that counting from 1 to n in binary takes $\Theta$(n) steps; i.e., the increment operation has constant amortized cost as opposed to $\Theta$(logn) in the worst-case. a) Analyze the amortized ...
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Which is the correct beginning to prove that $T$ is closed under $*$

Suppose that $*$ is an associative and commutative binary operation on a set $S$. Let $$T = \{a \in S \, : \, a*a = a\}$$ Is one of these methods or both a correct approach in beginning to solve ...
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Division by power of 3.

Is there any fast division algorithm to divide a binary number by power of $3$. I want to find the $q,r$ for $a=q*3^b+r$, $b$ is constant.
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3answers
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How to repeat a byte number inside another number without iterating? if possible…

Ok, I need a bit of help from my Math/Computer geeks out there. In the curse of an optimization for a program I am writing in Python, I found the following problem: for a given byte value, I need to ...
2
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2answers
36 views

Why in order to be a binary operation on $S$, each element of $S$ has to appear 'once and only once' in each row and column in Cayley Table?

I was reading about Composition table or Cayley Table; one of the points my book presents is that If all the entries of the table are elements of set $S$ and each element of $S$ appears once and ...
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Classes of binary operations between functions

Let $f,g : D\to \mathbb{R}$ be two functions defined from a domain $D\in \mathbb{R}$ to $\mathbb{R}$. I am looking for classes of binary operations $\circ$ between $f$ and $g$ that produce an $h:=f\...
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1answer
32 views

Decompose binary into decimal units, tens and hundreds

I have a 9 bit binary sequence (from 0 0000 0000 to 1 1111 1111) and I'd like to decompose into decimal units, tens and hundreds. Consider the following: 0 0111 1011 ==> 123 I'd like to ...
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2answers
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Finding unity and units of a binary operation

Let $\ast$ be the binary operation defined on $\mathbb{N}$ by $m \ast n = \max (m, n)$; the largest of $m$ and $n$. Decide whether unity exists and if so, find the units. I know that unity is defined ...
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Encoding and decoding with bitwise XOR and Shifts

This answer may exist somewhere already but if it does I've had trouble finding it. This is based on the problem from a programming site here $$encoded\_value(x) = { x \oplus (x<<1)|x, encoded\...