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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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
15 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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51 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
16 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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121 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
54 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
31 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
7 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
25 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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1answer
12 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
29 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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50 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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1answer
20 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
38 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
27 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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1answer
71 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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1answer
32 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
24 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
107 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
79 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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44 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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3answers
65 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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47 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
28 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
33 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
38 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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1answer
34 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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71 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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36 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
63 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
39 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)= ...
2
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2answers
132 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
70 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
143 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 ...
2
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1answer
35 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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4answers
100 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 ...
2
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3answers
104 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 $\mapsto$ ...
4
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1answer
44 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
33 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
35 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 = ...
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2answers
75 views

Prove that the group $(A,+, ◦) $ is a non-commutative ring

• $A × A → A, (f, g) → f + g$, where $(f + g)(x) = f(x) + g(x)$ for all $x ∈ K$ • $A × A → A, (f, g) → f ◦ g$ where $(f ◦ g)(x) = f(g(x))$ for all $x ∈ K$ Show that $(A,+,◦)$ is a non commutative ...
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125 views

What is subtraction?

Let $a, b \in \mathbf{R}$. It is an elementary fact that addition is a commutative binary operation on the reals, that is, $a + b \in \mathbf{R}$ and $a + b = b + a$. With the exception of ...
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1answer
22 views

Restricting Binary Operator $*$ To A Subset

I am currently reading Dummit and Foote's Abstract Algebra, and am having a little confusion over the following excerpt: Suppose that $*$ is a binary operation on a set $G$ and $H$ is a subset of ...
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72 views

How many associative ternary operations there are on a finite set?

We know that algebraic operation is a function $f:\underbrace{\left ( X\times X\times \cdots \times X\right )}_{t\ \text{times}}\rightarrow{X}$ If $X$ is a set and and cardinality is $|X|=n$ then ...
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1k views

How is addition different than multiplication?

Is there a fundamental difference in the things we call multiplication and those we call addition? In a field, both binary operations obey exactly the same rules (commutativity, associativity, ...
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159 views

Algebraically, What Does $\Bbb R$ get us?

In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$? Say we start with $\Bbb N$: ...
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83 views

Proving $K$ is a group

Now I have proved certain things are a group before, and I know that it requires: 1)Associativity 2)Inverse 3)Identity But here I have such a strange thing that I wanted to clarify that I am doing ...
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4answers
89 views

Show that * is associative

Could you show me how to prove the following to be associative? Please take me through the process step by step. $$a*b=a+b+2ab$$ Where $*$ is a binary operation and $a$ and $b$ are real numbers. I ...