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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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
39 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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27 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
32 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: ...
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3answers
46 views

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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1answer
15 views

Distributive law over addition

give example of a set with two binary operations addition and multiplication in which left distributive law holds but right distributive law does not hold i.e. a(b+c)=ab+ac but (b+c)a!=ba+ca or prove ...
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1answer
57 views

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
66 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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57 views

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 $x≥1$, ...
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14 views

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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25 views

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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1answer
17 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
18 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 ...
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26 views

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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56 views

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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0answers
15 views

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 ...
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3answers
34 views

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
19 views

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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1answer
17 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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16 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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28 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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53 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 ...
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119 views

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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1answer
22 views

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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28 views

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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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 ...
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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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0answers
19 views

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 ...
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1answer
28 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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29 views

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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0answers
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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, ...
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1answer
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Binary operation - finding identity

Here is the question: The binary operation * is defined for $x,y \in\ S = {0,1,2,3,4,5,6}$ by $x*y=(x^3y-xy)mod7$ Find the element $e$ such that $e*y=y$ for all $y\in\ S$ So far I have the ...
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1answer
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Mathematical Expression for multiplication and addition of binary arrays

Assuming I have two binar arrays with 3 bits. Array A = 1,1,0 and Array B = 0,1,1. Now I want to perform bit-wise AND operation ...
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Group Theory and Cardinals [duplicate]

I was wondering the following: Let's suppose that $G$ is a non-empty set, then, can we always find a binary operation $*$ such that $(G,*)$ is a group? For example, if we fix $G=\mathbb{Q}$ the sum ...
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67 views

Right way to show associativity.

Authors usually write that $*$ is associative on a set $S$ if, $(a*b)*c=a*(b*c)$ $\forall a,b,c \in S$ I think it should have been, $(a*b)*c=a*(b*c)=(a*c)*b$ $\forall a,b,c \in S$ I made all ...
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2answers
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Find a fraction given the repeating binary expansions

I can do binary expansion given a fraction just fine but the question I'm stuck on says: Find fractions for the numbers with the following binary expansion: (i) $0.00\overline{110}$ and (ii) ...
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10answers
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Non-associative commutative binary operation [duplicate]

Is there an example of a non-associative, commutative binary operation? What about a non-associative, commutative binary operation with identity and inverses? The only example of a non-associative ...
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2answers
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With 8 bits is it possible to obtain an integer in more than one way? [duplicate]

This is just a curiosity that just came to my mind while thinking at IP addresses. A byte is composed of 8 bits. A bit can either be $0$ or $1$. IPv4 addresses are composed of a group of 4 bytes. ...
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Name of this property: if $x * x = y * y \implies x = y$

Algebraically speaking, what's the name of this property?: $x * x = y * y \implies x = y$ $*$ being a binary operation
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Symbol for “the greater of the two values”

I'm looking for an operator that returns the greater of two values. Here's an example. If $a=5$, $b=6$ and $???$ is the operator, I'd like to have $x$ equal $b$ when I do $x=a???b$, since $b$ is the ...
4
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Properties and geometrical interpretations of a specific planar vector law.

Genesis of the following thoughts. One usually define the following internal composition law on $\mathbb{R}^2$: $$+:\left\{\begin{array}{ccc} ...
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2answers
95 views

efficient way to express large numbers

I recently watched the walkthrough of Graham's Number on YouTube (Numberphile). Mind-blowing of course. I then puttered around in other large number topics like Ackerman and Tree(3) and fast growing ...
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1answer
289 views

Are the addition and multiplication of real numbers, as we know them, unique?

After recently concluding my Real Analysis course in this semester I got the following question bugging me: Is the canonical operation of addition on real numbers unique? Otherwise: Can we define ...
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8 views

How would I round the number $-(1.01101|01)_2$ down?

I'm trying to round $-(1.01101|01)_2$ in binary down. The $|01$ are the guard bits. The answer is $-(1.01110)_2$ My attempt is as follows: $-(1.01101|01)_2$ $-(0.00001|00)$ = $(1.01100)$ Not ...
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A name for this property?

Let $*$ be an operation such that $(xy)^* = y^*x^*$, e.g. if $x,y$ are $2\times2$ matrices and $*$ is "take the inverse" or if $x,y$ are operators and if $*$ is the adjoint. Is there a name for such ...
2
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1answer
36 views

Prove the following set endowed with the binary operation is an abelian group

Let $∗$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x ∗ z = y$. Show that this set with the operation ...