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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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Having trouble determining what Turing machine evaluates?

While learning about Turing machines, I've stumbled onto a problem that I'm not sure how to solve. I've put a lot of work into trying to find the solution, any help would be appreciated. The problem ...
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0answers
19 views

How to divide two binary number in the 2's complement form? [closed]

Let's say : divide $10001000_b$ by $00100010_b$ in the 2's complement form. $_b$ denotes binary.
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1answer
52 views

Is * an example of a binary operation?

I think I understand what a binary operation is, but I want to check my understanding with this example to be sure. First of all, my textbook defines a binary operation in the following way: Let $A$ ...
0
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2answers
25 views

Binary matrix multiplication: finding the number of ones [duplicate]

Consider a binary matrix $\mathbf A_n$ corresponding to values $0$ to $2^n-1$ where each row represents a length $n$ binary representation of a real number. For example, for $n=3$ we have $\mathbf ...
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0answers
13 views

Number of ones in a binary matrix multiplication with a vector

Consider a binary matrix $\mathbf A$ corresponding to values $0$ to $2^n-1$ where each row represents a length $n$ binary representation of a real number. For example, for $n=3$ we have $\mathbf A=\...
0
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1answer
30 views

Find the binary of decimal numbers given with powers of 10 [closed]

Convert to binary : 46.5 * 10^(-24) Or something like 46.5 * 10^(24) I have to find the binary equivalents here, for the purpose of representation in IEEE 754 floating point representation. But I ...
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0answers
12 views

Clarification about binary bijection

I'm struggling with a sentence said during a lecture.. There is something that I can't get, maybe due to a misunderstanding in the notation.. that I have never used before. "Let's consider the ...
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1answer
37 views

Prove the following for a general binary operation

I need some help with the following proof, please. First, a definition below: A binary operation $p$ on a set $X$ is a function of two variables, whose values lie in $X$: it assigns to each ordered ...
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2answers
27 views

Is there a notation for iterated/repeat concatenation?

Given a string x and natural number y, is there a commonly used notation for a function that concatenates string x to itself y times? Example: $x = \mathrm{'foobar'}$ $y = 3$ $f(x,y)=\mathrm{'...
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1answer
43 views

I need to solve one equation, but I dont know how to solve equation with floor functions

Im a student with not such a knowledge to solve equations with floor functions. I want to ask, if it is even possible and if it so, how is possible to prove this equation to be true. ...
0
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1answer
33 views

How to prove that an operation is binary?

I have been trying to learn binary operations and have not been able to understand how to prove that an operation is binary. For example: Show that $*:\mathbb R×\mathbb R→\mathbb R$ given by $(a,b)\...
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2answers
36 views

Is there a name for a boolean operator whose value always equals its left-hand side operand?

Given a binary boolean function with truth table as follows: ...
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1answer
40 views

Definition of Additive Loop

I have thought of loops as having the operations of multiplication and left and right division. I read the D. R. Hughes article on Additive and Multiplicative Loops of Planar Ternary Rings and it ...
2
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1answer
47 views

Abelian groups about rationals [duplicate]

Is the set $\mathbb{Q}$ under $×$ an abelian group? It is sure for $\mathbb{Q} - {0}$, but i think the whole set of rationals is not an abelian group as $0 × a = a × 0 = 0$, but the identity element ...
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1answer
23 views

Is there a sum-like operator over subsets of 1 or 2 elements over $\mathbb{Z}$

Let be $A(\mathbb{Z})$ the set of subsets of 1 or 2 elements, like: $\{ 1 \}, \{ 1, 2 \}$. I would like to know if we could prove there is no map $+ : A(\mathbb{Z}) \times A(\mathbb{Z}) \to A(\mathbb{...
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1answer
28 views

Abstract Algebra - Binary Operation [closed]

Determine whether the definition of * does give a binary operation on the set. If it is a binary operation, determine whether if is commutative and associative. On Z, define * by a*b=a^b
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2answers
46 views

How to prove formally that division is not commutative?

I am studying quantitative reasoning and I am wondering how to prove formally that division is not commutative. Everyone knows that 7/3 is not the same as 3/7, but is there a formal way to ...
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2answers
31 views

Need help confirming which algebraic structure this is

Let's define a binary operation $*$ on $\mathbb{R}$ such as $$ a * b = e^{a+b} $$ and investigate which algebraic structure this is. Well first of all we notice that the operation is closed under $\...
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1answer
21 views

A set operation is associative if and only if the binary operator defining it is associative

Let $\star$ be a binary operation on $\Bbb{N}$. For all $A,B \subseteq\Bbb{N}$, $\circ$ is defined as: $$A \circ B = \left\{a \star b \mid a \in A \wedge b \in B\right\}$$ I'm trying to prove that $...
2
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0answers
98 views

How many distinct subsets of binary boolean operators are closed under composition?

Question: There are $2^4=16$ distinct binary boolean operators. Two operators are regarded the same if one can be obtained from the other by exchanging the operands (input). It is easy to see only $...
0
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1answer
30 views

How to choose a set of boolean functions with a specified probability of getting a 1

So let's say I have a boolean function $f(x)$ that takes in a size k binary vector and outputs a binary scalar. Each function is defined as a $2^k$ vector. For example $f((0,0)) = 0, f((0,1)) = 1, f((...
2
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2answers
35 views

What is the significance of the precedence of a semiring/ ring operations?

In general, a semiring/ring is equipped with two distinct binary operations, namely addition $(+)$ and multiplication $(×)$ , where in most of the cases, multiplication distributes over addition and ...
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0answers
32 views

Does this algebraic structure belong to a subclass of neofield (or other structure)?

This article indicates that fields may be subclasses of neofields to the extent that they are neofields with associative addition. The author uses the neofield structure exclusively with finite sets, ...
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4answers
107 views

Should we distinguish the minus sign from the negative sign?

In the set $\mathbb{C}$ of complex numbers, the minus sign "-" may be used for following: As a unary operator $-_u$, given a complex number $a$, $-_ua$ is the unique number (called the negative of a) ...
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1answer
47 views

Is $a*b = a - b + 1$ a binary operation on $\mathbb{Z}^+$?

Possible binary operation is $a*b = a - b + 1$ for all $a,b \in \mathbb{Z}^+$ I don't believe this is a binary operation in any way. A counter example is when $a = 1$ and $b = 5$, then the result of $...
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2answers
28 views

Showing associativity on this binary operation

I am trying to determine whether the definition of * gives a binary operation on the set. On $\mathbb{Z^+}$, define * by letting $a*b = a^b$ I think that the binary operation is not commutative ...
2
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3answers
92 views

Is an inverse element of binary operation unique? If yes then how?

I am trying to prove it but not getting any clue how to start it! $$a*b=b*a=e,$$ $$a*c=c*a=e$$ How to show $b=c$?
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0answers
30 views

Solving for a and b from xor and difference

Can we solve for a and b, from $a-b = x$ $a \oplus b = y$ Even when x and y can be negative? I tried substituting the value of a in xor equation from the ...
1
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2answers
35 views

Binary Relations - Definition

I am familiar with the definition of a binary relation from set $A$ to set $B$ as a subset of their Cartesian product $A × B$. I do not understand, however, how one can view certain mathematical ...
2
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1answer
24 views

How to show a piecewise binary operation over functions is associative?

Given some set of functions $S(X) = \{f:[0,a]\to X : f(0)=f(a)=x_0 , a \geq 0\}$, define the binary operator $*$ over two functions $f:[0,a]\to X$ and $g:[0,b]\to X$ as: $$(f * g)(t) = \begin{cases} f(...
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1answer
55 views

Subtraction With 8 Bit Integers

I have encountered the following question and I don't know how to approach it: "Perform the following subtraction by adding the 2's complement using 8 bit integers: 35-15=20"
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1answer
28 views

Encoding numbers from 0 to 255 using Huffman coding.

How can I encode numbers from 0 to 255 using Huffman coding (or any other code), so that each number (especially the largest numbers such as 255) wouldn't take 8 bits of binary space? In other words, ...
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2answers
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Stuck in something: How to write coordinates in one number?

I have a X, Y coordinate system which starts from 0 and ends in 255 on each axis. Thus, I can fit 65,025 numbers in it. Imagine each number as a pixel, so I have 65,0250 pixels in my coordinate system....
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0answers
13 views

Name of binary operation property to be equal to constant on equal arguments

How can someone call property for some binary operation $f\colon X\times X \to X$ that $$ \exists a\in X: \forall x \in X \quad f(x,x) = a $$ ? Examples are $f(x, y) = x-y$ and $a = 0$. $f(x, y) = ...
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4answers
221 views

Do isomorphic groups share a binary operation? [closed]

Suppose you have two isomorphic groups. Does the binary operation defined on each group need to be the same operation?
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1answer
22 views

Why we can double elements in the table?

If we perform an operation on ordered pair from set A={1,2,3,4}, we can built a table like this: But, since set A is finite and there is only one element "1", "2" etc., why we can double elements in ...
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2answers
33 views

Series of binary operations in $(\mathbb{N}, *)$

Consider monoid $(\mathbb{N},*)$, where operation $*$ is defined as $x*y = xy + x + y$. What is the result of $1*2*3*\text{...}*25 \text{ (mod 29)}$ ? $xy + x + y$ can be written as $(x + 1) y +...
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1answer
22 views

Find the binary input function given the outputs (part 2)

Here we have three binary variables $x_1$, $x_2$, $x_3$ $\in \{0,1\}$. I want to find the form of the function $f(x_1, x_2, x_3)$ such that the following are satisfied: if $\ x_1 = 0,\ x_2 = 0,\ ...
10
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1answer
129 views

Prove that a semigroup satisfying $a^pb^q=ba$ is commutative

Let $(S, \cdot)$ be a semigroup. There are natural numbers $p,q \geq 2$ such that $a^pb^q=ba$ for all $a,b \in S$. Prove that $S$ is commutative. I wrote $$\begin{align} a^{p+1}b^{q+1} &=b^{(q+...
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1answer
42 views

How to prove that a relation is transitive when the binary operation is commutative?

Let $A$ be a non-empty set and suppose that $*$ is a binary operation on $A$. We define a relation $R$ on the set $A$ as follows: $R = \{ (x,y) \in A \times A: x*y=y*x\}$. In other words, given $x,y ...
1
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1answer
27 views

Known Algebraic Structure(s) by Operation and Element Characteristics

Is there a known algebraic structure with two binary operations defined such that one of the operations behaves like multiplication (identity, distributive, commutative, and associative) and the other ...
3
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2answers
215 views

Does invertability and closure imply identity?

Sorry if this is a basic question or I'm overthinking it, but if an algebraic structure has inverse elements (or at least for a member $a$), that means $a^{-1}a=e$, and if there's closure then e is an ...
19
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1answer
1k views

A question from 1989 leningrad mathematical olympiad

Prove that we cannot define an binary operation $*$ on the set of integers Z satisfy all of the three properties below simultaneously: For any $A∈Z,B∈Z,C∈Z:$ 1.$A*B=-(B*A)$ 2.$(A*B)*C=A*(B*C)$ (...
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2answers
38 views

Find the binary input function given the outputs

Here we have three binary variables $x_1$, $x_2$, $x_3$ $\in \{0,1\}$. I want to find the form of the function $f(x_1, x_2, x_3)$ such that the following are satisfied: if $\ x_1 = 0,\ x_2 = 0,\ ...
1
vote
1answer
30 views

determine whether this operation is binary

Define the operation $*$ on the set $M_2(\mathbb{Z})$ as: $A*B = AB+aBA$. Determine $a \in \mathbb{R}$ such that $*$ is binary. My attempt: \begin{bmatrix} z_1w_1+z_2w_3+aw_1z_1+aw_2z_3 & z_1w_2+...
0
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1answer
68 views

How to prove $A=R-\{-1\}$ and $a*b = a+b+ab $ is a binary operation?

$A=R-\{-1\}$ and $a*b = a+b+ab $ Show that * is a binary operation on A Show that * is associative Show that there is an identity element in A for * Show that every element in A has an inverse with ...
2
votes
2answers
183 views

Find elements from xor relations

Alice and Bob are playing a game. Alice has a sequence of positive integers $$a_1,a_2, \ldots, a_N;$$ Bob should find the values of all elements of this sequence. Bob may ask Alice at most $N$ ...
1
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1answer
37 views

Concatenation operation on the set of finite sequences in $\{0, 1\}$

Let $A^{\ast} = \bigcup_{I \subset \mathbb{N}} \mathcal{F}(I, \{0, 1\}) = \bigcup_{I \subset \mathbb{N}} (\prod_{i \in I} \{0, 1\})$ be the set of finite sequences in $\{0, 1\}$. First, if $I = \...
6
votes
2answers
125 views

Is the division symbol $\div$ acceptable based on international standards? [closed]

The division symbol $\div$ is found in almost all calculators; however, I seldom see it in any formal writing. It seems people almost exclusively prefer $\frac{a}{b}$, $a/b$ or $ab^{-1}$ to $a\div b$. ...
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0answers
13 views

Question about order of operations convention and alternative

I'm (trying) to make my own code parser that evaluates expressions. I've done some reading about this topic and seen the various mnemonics that school children are taught, such as: P(arentheses) E(...