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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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Abstract Algebra - Binary Operation [on hold]

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 ...
2
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2answers
28 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 $\...
0
votes
1answer
20 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
94 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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0answers
26 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
votes
2answers
32 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 ...
0
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0answers
30 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, ...
9
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4answers
95 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) ...
1
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1answer
46 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 $...
0
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2answers
26 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
votes
3answers
85 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
29 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
33 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
votes
1answer
23 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(...
0
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1answer
35 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"
0
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1answer
24 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
37 views

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....
0
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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
217 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
17 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 ...
2
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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 +...
0
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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
126 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+...
0
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1answer
36 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
24 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
votes
2answers
210 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.$(A*B)*C=A*(B*C)$ (...
0
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2answers
37 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
27 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
38 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
167 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 $N$ ...
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1answer
34 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
123 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
12 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(...
0
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1answer
19 views

a conjecture on the binary operation of multiplication [duplicate]

Give a conjecture describing the values of $n$ for which all of the nonzero elements of $Z_n = {0, 1, 2, . . . , n − 1}$ have multiplicative inverses. I am guessing the point of not having $0$ ...
0
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1answer
92 views

A well-defined binary operation on a class of functions (Eudoxus magnitudes) from $\mathbb N$ to $\mathbb N$?

Update: The answer shows that some tweaking is necessary to get this to work. The problem are those $f$ where there exist an $N$ such that $f(n)$ is always odd for $n \ge N$. But this can be remedied ...
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3answers
175 views

Is there ANY context in which f(x,x) is noncommutative?

Stupid question, but one occasionally reads such things as "the operation $\ast$ is noncommutative for all $x,y$ such that $x\neq y$" or "$x\ast y$ is commutative iff $x=y$". These statements bother ...
7
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7answers
2k views

Why should a non-commutative operation even be called “multiplication”?

As per my knowledge and what was taught in school, $a\times b$ is $a$ times $b$ or $b$ times $a$ Obviously this is commutative as $a$ times $b$ and $b$ times $a$ are same thing. On the other ...
0
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1answer
25 views

Bounds of defining operations for equivalence classes

When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific ...
4
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2answers
75 views

Is there a set in which division of 0 by 0 is defined?

The reason I ask this is that I've discovered that, even though they don't satisfy all field axioms, there are sets called projectively extended real number line and Riemann sphere, which are ℝ∪{∞} ...
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0answers
29 views

Function/Operation Definitions

If I want to define a function/operation for equivalence classes, is it permissible to stipulate contingencies based upon element characteristics or does dependency on the choice of element ...
6
votes
2answers
238 views

An $m$-ary function that represents all $n$-ary functions

Motivation It is well-known that any binary operator $*$ on the boolean ring $\{0,1\}$ can be represented using only one of the $\operatorname{NAND}$ and $\operatorname{NOR}$ operators. For example,...
2
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1answer
43 views

Definition of “fixed point” for binary operator

What is a term for a value $x \in X$ that for binary operator $f\colon X\times X \to X$ maps always to itself, no matter what is the other value $$ \forall y \in X \quad f(x, y) = f(y, x) = x $$ Is ...
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0answers
37 views

Help Me Understand the Generalisations of Commutativity and Associativity for Unions

Introduction I can prove the generalisation of commutativity and associativity of unions, but I do not understand how they are generalisations of commutativity and associativity. I think it would ...
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2answers
40 views

Inverse operation of xor

If x = a xor b, given the values of x and a can we find b? In other words, which function can be applied on both sides in the equation to get the value of b?
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0answers
32 views

Floating Point to Decimal Conversion

If a floating-point number is stored in one byte such that the first bit is the sign, the next three bit represent the exponent in excess-4 notation, and the last four bits represent the mantissa, ...
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1answer
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Is $f(g,h) = g^2$ a binary operation on a group $G$? [closed]

If $(G,\cdot)$ is a group, then $f : G \times G\to G$, defined by $f(g, h) = g^2$, is a binary operation on $G$. Is it true or false and why?
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1answer
25 views

Image of a binary relation

I know it might seem a stupid question but I need some clarification. From my notes: "a binary relation $R$ associated to each element $x$ of $X$ some elements $y$ of $X$. We denote by $R(x) = \{y \...
0
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1answer
47 views

If $x,y \in \mathbb{N}$ then $x+y=0 \iff x=y=0$

Let $x,y \in \mathbb{N}$. The operation $(+)$ is defined by: $$x+0=x$$ $$ x+(y+1)=(x+y)+1$$ Then prove that $x+y=0 \iff x=y=0$. The second implication $x=y=0 \implies x+y=0 $ is simple and ...