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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How to expand powers of multiple pairwise commuting elements in a group [on hold]

Let (G, $\ast$) be a group and let n $\in\aleph$. Prove that if $g_1,...,g_k\in G, k\in\aleph$ are pairwise commuting elements of G, then $(g_1\ast...\ast g_k)^n$=$g_1^n\ast ...\ast g_k^n$
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Prove that the following set is a group

Prove that that the following is or is not a group. (a) The set S = $\mathbb{R}$ \ {0} with operation defined by a * b = 2ab for all a and b in S. (On the right side of the equation, the operations ...
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Binary operation $xy$, has identity, but not associativity. Is the inverse unique?

Let $S$ be a set with a binary operation $xy$ defined on it, with a neutral element, but not satisfying associativity. I want to prove that the inverse isn't necessarily unique. My attempt to answer ...
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1answer
63 views

Strange binary operation: $a\oplus b = (a-b)||(a+b)$

Motivation: A facebook post had a bunch of these as an 'intelligence' test, so I thought I would think about what this operation is. But I haven't done this Math in years! I have an operation ...
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1answer
71 views

Find number between $A$ and $B$ with maximum set bits?

Given two integers $A,B$. Find number $N$ which has maximum number of set bits in its binary form and lies between $A$ and $B$ inclusive. Is there any approach for this question. Also if there are ...
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59 views

Using XOR operation repeatedly

There are $n$ binary digits, from $A_0$ to $A_{n-1}$. Each operation consists of the following 2 steps: Each digit is replaced by the XOR addition of itself with the next digit. ...
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1answer
61 views

Is distributivity sufficient to define composition?

Function Composition has the property of distributivity: $$(f\star g)\circ h = (f\circ h)\star(g\circ h)\;\forall f,g,\star \in\{+,-,\times,\div\}$$ I was wondering if these properties uniquely ...
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26 views

How many numbers with given amount of ones in their binary form?

I was practicing for a programming competition and I got the following problem, which I was unable to solve: It is given a number N. Find the amount of x, y values, where x > N, y < N and the ...
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2answers
115 views

binary subtraction

I am trying to solve binary subtraction: $$11000_2 - 1011_2 = 1001_2$$ I know that it should be $1001_2$, when checking with answer key however I am not sure how it was calculated as I get different ...
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1answer
18 views

float vector to binary integer vector transformation preserving dot product

Is there a transformation of a set of float vectors to a set of binary integer vectors that preserves the dot product. I found conformal transformations but I'm interested in large vectors (size 300) ...
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68 views

How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
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1answer
19 views

Binary Subtraction with negative result

I want to do this little subtraction (but with bits): $1372 - 9714$ The binary code I found for $1372$ is: $00010101011100$ The binary code I found for $9714$ is: $10010111110010$ Then I added a ...
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137 views

Binary operation commutative, associative, and distributive over multiplication

Is there any binary operation that is commutative, associative, and distributive over multiplication? I asked this question in my head a while ago, and I posted it in various forums. However, having ...
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1answer
58 views

Can a smooth function on the reals form a non-commutative semigroup?

Let $f\colon \mathbb{R}^2 \to\mathbb{R} $ be a smooth function. Can there exist an algebraic structure $(\mathbb{R}, \cdot)$ such that for $x,y \in \mathbb{R}$, $x \cdot y = f(x,y)$ that is a ...
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1answer
34 views

How do I complete this “Cayley table” or binary operation table?

I have an algebraic structure $(S,\cdot)$ and $a,b,c,d \in S$ where $a,b,c,d$ are not necessarily four distinct elements. This is part of a larger problem that I am working on and based on what I ...
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0answers
16 views

Converting a boolean expression into CNF and DNF

Is there any systematic way to convert the following boolean expression (QUBO) into CNF or DNF? Here, $x_1, \ldots, x_n \in \{0, 1\}$, $a_1, \ldots, a_n \in \mathbb{Z}$ and $b_{1,1}, \ldots, b_{n,n} ...
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1answer
29 views

how to find the binary expansion of any number in the unit interval [0,1]

For each integer $n\geq 1$ and $x\in [0,1]$, define $f_n(x)=x_n$ where $x_n$ is the $n$th binary digit of x. If x is a number with two binary expansions, use the expansion that ends with infinitely ...
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13 views

Mathematical Terms for Binary Operations

I'm trying to represent binary operations on numbers in mathematically correct terminology. For example given two binary numbers: 42 : 101010 13 : 001101 I want ...
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1answer
30 views

Distributivity of $\times$ over $+$

I had a problem recently as part of class work that dealt with sets and the conditions imposed by them. This is part of a larger question that I simplified things down to. The parts I'm concerned with ...
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56 views

Association, Commutation and Identity Elements on Binary Operations?

Is the following closed, associative or commutative? f(a, b) = (a+b)/2, where a, b ∈ Z. I found that it is not closed but I am not sure how to find whether or not it is associative (I was confused ...
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25 views

Which of the following sets are closed under the binary operation $*$ defined as:$a*b=\frac{a+b}{1+ab}$

Which of the following sets are closed under the binary operation $*$ defined as: $$a*b=\frac{a+b}{1+ab}$$ $1.\{x\in \mathbb{R}:x\geq 0\}$ $2.\{x\in \mathbb{R}:|x|>1 \}$ $3.\{x\in ...
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1answer
40 views

Number of different magmas up to isomorphism

Let $(M,\circ)$ be a magma over a finite set of order $n$. I tried to count all the possible magmas up to isomorphism, but I just can't get it right. My naive approach was to count all the possible ...
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1answer
29 views

Transitive Relations on a set

I am trying to study binary relations (for myself, it's not an assignment!) I have the set $\{1,2,3,4\}$, and one of the relations in the exercise is $\{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)\}$. A ...
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74 views

How to define a taxonomy of non associative operations?

Let $A$ be a set, and let $a,b,c\in A$. Let also $\circ: A\times A\rightarrow A$ be a binary operation on $A$. We agree as usual to write $a\circ b$ to mean $\circ(a,b)$. We say that $\circ$ is ...
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33 views

Where can I find a binary calculator that can do exponentiations, roots and logarithms?

I've searched on Google, but all I found was binary calculators that can do additions, subtractions, multiplications and divisions, nothing else.
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1answer
28 views

Express a binary operation in decimal

Is there a way to represent binary operation in decimal. What I mean with this is for example a set of decimal operators that would give the same result as a x>>n a ror(x), etc. So far the only thing ...
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1answer
121 views

Absolute Value Equivalence relation inequality Question

I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me? For $a, b ...
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1answer
90 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
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45 views

Decimal binary sequences that cannot be greater than $1$

Consider the family of sequences of the form $.012\ldots n$ for any natural number $n$. So, the sequences in this family are: $.01, .012, .0123, .01234,$ etc. Now consider to manipulate each ...
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67 views

Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
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49 views

Does the binary operation $m ⋆ n = m^n$ on $\mathbb N$ have a neutral element?

Does the binary operation $\,m ⋆ n = m^n\,$ on $\,\mathbb N\,$ have a neutral element? I said yes, and it is $\,e=1\,$ because $\,m ⋆ e = m^e = m^1 = m,\;$ but apparently that is wrong.
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Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
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1answer
32 views

How to compute associative binary operation on a finite set based on partial information?

I am working on a problem, and I must be staring at the answer without seeing it since it's among the introductory problems in my abstract algebra textbook. We're told that an associative binary ...
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1answer
34 views

Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
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1answer
42 views

Associative law with negative sign

We know that $a+b+c$ is meaningful for, say, the group $\left( \mathbb{Z}% ,+\right) $. Since for any $b,c$, we must have $b^{-1},c^{-1}$, therefore $% a+b^{-1}+c^{-1}$ has to be meaningful, too, but ...
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35 views

How do I prove (scalar1 + vector1) * scalar2 is not equal to scalar1 * scalar2 + scalar2 * vector1?

I am taking a Linear Algebra course and have been stumped on a homework question for a few hours. How do I prove for two scalars, $c_1$ and $c_2$, and a vector $v$: $(c_1 + v)c_2 \neq c_1c_2 + c_2v$ ...
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72 views

Is this binary operation a group?

Let $Y=\left\{(a,b)\in\mathbb{R}\times\mathbb{R}\ |\ a\ne 0\right\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)∗(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$. I already did the ...
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41 views

how to prove this group of the binary operation

Let $Y=\{(a,b)∈ \Bbb R\times \Bbb R∣ a≠0\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)*(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$. I already do the proof of ∗ is an operation on ...
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2answers
77 views

How to determine if a set is closed under some operation?

Is the set $\{-2,0,2\}$ closed under addition? And why? Specifically, when determining if a set is closed under an operation do you apply the operation to the each number and itself? For ...
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1answer
45 views

How to prove that: If two binary operations are anti-isomorphic and one of them is associative then the second one also will be associative?

We know what is called an anti-isomorphic operation on a set S. it is just a one two one $ g $ function mapping from $S$ to $S$. $ g: S \rightarrow S$. and it satisfy this condition $ g(xy)= ...
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138 views

Algebraic Structure: Are Set Operations Considered Binary Operations?

I'm currently trying to understand the "hierarchy" of sets / algebraic structures, e.g. things like groups, rings, fields, modules, algebra, vector spaces which I mostly understand, but especially the ...
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1answer
74 views

Clarify Cartesian Products and Binary Operations

So tell me if I'm saying this write. A Cartesian Product is a function f:X x Y --> Z , where some unknown structural operation on the sets X and Y produces a set Z as its codomain, and Z is a set of ...
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1answer
159 views

For an associative binary operation with identity, the set of invertible elements forms a group

Let $S$ be a set, and $*$ an associative binary operation on $S$. Suppose there is an element $e\in S$ such that ($1$) $e*x=x$ and $x*e=x$ for all $x\in S$. (a) Prove that there is a unique element ...
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1answer
22 views

How to create a new binary operation on a same set?

I was studying binary operation on a set. Then the following question came to mind. I tried to find an answer. also searched in website but could not get any satisfactory answer. the question is: is ...
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1answer
38 views

Associative Numbers

Yesterday my friend wrote a number on a paper. He then added the number of ones in the binary representation of the number to that number and formed a new number. He kept doing the process infinitely. ...
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101 views

Explaining multiplication of fractions

The best way I've been able to describe multiplication is as $$ a\times b = \sum^a_{i=1} b$$ But my definition does not account for things such as $2.99792458\times8.987551787$ and ...
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3answers
107 views

What is a bit-shifting standard C function for calculating $f(x) = \frac{(2^{16}- 1)}{(2^{32} - 1)}\cdot x$

I need to take 32-bit unsigned integers and scale them to 16-bit unsigned integers "evenly" so that $0 \mapsto 0$ and 0xFFFFFFFF $\mapsto$ ...
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49 views

Is the number of associative $n$-ary algebraic operations on a finite set with 2 cardinality always 8?

We know that if $n = 2$ then the operation is called a binary operation. $ \circ $ on set $X$ is a function $\circ : X \times X \rightarrow X$. And the number of all associative binary operation on a ...
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1answer
35 views

Reducing Several Identities to One Identity

One class of algebraic structures that are typically studied are those given by a set $X$ and a set of $n$-ary operations defined on $X$ for each $n\in \mathbb{N}$. Perhaps most studied are those ...
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1answer
39 views

Multiplication by scalar axioms for an abelian group.

There is an R vector space where $k ⊙ x := x^k$ , $∀x, y ∈ V, k ∈ R$, I showed that it was abelian. I wanted to show scalar multiplication by using the axioms. $α ⊙ (x ⊕ y) = α ⊙ (xy) = (xy)^α = ...