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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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Exist a way to return a unary operation for algebraic structures? [on hold]

I read from here The essence of an “operation" is that two things are combined to form a third thing of thesame kind A binary operation on a set G is a function $∗: G×G \rightarrow G$ This ...
0
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1answer
20 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,\ ...
0
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0answers
16 views

Can we re-read natural numbers as a “binary operation” of Integers Numbers? [closed]

I start from Binary Operator definition I see an affinity between this definition and the 'difference' between Integers and Natural numbers. I ask if I can value in this way +1 -1 as $x, y$ |1| as $...
9
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1answer
107 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
27 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
23 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
203 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
32 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
24 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
24 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
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2answers
152 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
32 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
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2answers
118 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
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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
18 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
90 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 ...
1
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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 ...
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7answers
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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
73 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
27 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
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2answers
231 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,...
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0answers
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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 ...
0
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2answers
24 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?
0
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0answers
25 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
40 views

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?
0
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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 \...
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1answer
46 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 ...
4
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2answers
117 views

Commutative Semigroup

Let $S$ be a Semigroup with the two following properties, $(1):$ for all $x$ in $S$ we have $x^3=x$ $(2):$ for any $x,y$ in $S$ we have $xy^2x=yx^2y$. Then prove that this Semigroup $S$ is ...
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1answer
43 views

Check the properties of the following operation defined on R

An operation is defined on $\mathbb{R}$ such that for every $x,y \in \mathbb{R}$, $x \ast y=\sqrt{x^2+y^2}$. I was checked some of the basic properties like commutativity, associativity and whether ...
0
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1answer
47 views

Is it possible to reduce an algebraic function to $1$s and $0$s?

I want to know where in this process I'm going wrong. Perhaps it's not even a valid thing to do...? Take a well-behaved function such as $f(x)=x \sin 2x$. I want to turn this into a new function $g$ ...
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3answers
38 views

Set $S= \mathbb{Q} \times \mathbb{Q}^{*}$ with the binary operation $(i,j)\star (v,w)=(iw+v, jw)$

Consider the set $S= \mathbb{Q} \times \mathbb{Q}^{*}$ with the binary operation $(i,j)\star (v,w)=(iw+v, jw)$. (a) Show that the binary operation is closed. I said the operation is closed under ...
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1answer
36 views

isomorphism in a product

$x*y=\frac{x+y}{1+xy} , x,y\in(-1,1).$ Calculate the value of $ \frac{1}{2}*\frac{1}{3}* \cdots *\frac{1}{1000}.$ I tried a lot of functions but I don't know how to find a good isomorphism and do the ...
1
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1answer
49 views

Ambiguity with parentheses multiplications [closed]

I was recently shown the equation $6 \div 2(1 + 2) = ?$, and it was disputed whether this equation equals $1$ or $9$. To solve for $1$: $$ 6 \div 2(1 + 2) \\ 6 \div 2(3) \\ 6 \div 6 \\ 1 $$ To ...
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0answers
20 views

Proper class of operations

Usually, a collection of operations forms a set. But I've heard of an example where it is (must be) a proper class. Namely, that $\mathbb{CompHauss}$ becomes a variety if proper class of operations ...
4
votes
1answer
91 views

Name for generalization of property: $f^n(x) \ne x$ for all $n > 0$

I am curious about how to specify with standard terminology that a certain function is non-repeating, in the following sense: In the simple case of a unary operation $f: X \to X$, this property would ...
0
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1answer
26 views

Which integers have inverses with respect to lcm operator, a*b=lcm (a,b) [closed]

I know that the identity of this operator is 1, but am not sure about the inverse.
1
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1answer
94 views

If symmetric difference leaves a set unaffected, the second set is empty.

To prove: If $A \triangle B=A$, then $ B= \emptyset$. This seems simple enough, as an idea. I mean if the set $B$ is anything but empty, $A\triangle B$ would contain more or less than simply $A$, ...
14
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4answers
1k views

Do binary operations need to be surjective functions?

Let $\star$ be a binary operation on the set $S=[0,1]$ defined to be $$\star : [0,1] \times [0,1] \to [0,1] $$ $$\text{where } a \star b = \text{min}\left(\frac12 a , \frac12 b\right) $$ From ...
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0answers
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Binary operation of addition on reals is uniformly continuous, whereas multiplication is not (proof-verification)

I would like to ask whether the following proof is correct and, in particular, why the step (*) of this proof is valid? The problem is to show that the function $+: \mathbb{R}^2 \to \mathbb{R}, (x,y)\...
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0answers
32 views

How to find the number of msb bits common between two binary numbers

Im trying to find the number of bits common between two binary numbers starting from MSB -> LSB. For example, I'm taking a set of binary numbers of 4 bits each i.e., 0 -> 15. I'm trying to find the ...
0
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0answers
34 views

How define formally a user-defined binary operation?

Let $x \in \{0,1\}^{|\mathcal{S}|}$ and $k \in \{0,1\}^{|\mathcal{R}|}$, where their elements are indexed by the index sets $\mathcal{S}$ and $\mathcal{R}$ respectively. The index sets satisfy $\...
7
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0answers
170 views

Associative, non-commutative, non-trivial, analytic binary operation

There was a question whether associative, but non-commutative binary operation over the real numbers exist. A trivial answer is the binary operation $x\circ y = x$ or $x\circ y = y$. As a followup, ...
8
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2answers
226 views

Associative, non-commutative, nontrivial operation on the real numbers

This MSE question asks about binary operations on the real numbers which are associative, but not commutative. Two answers are given: The operation $\circ$ defined by $x \circ y=x$. Letting $f:\...
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0answers
39 views

Generalised Binary Operation?

In algebra, we study sets with binary operations satisfying certain properties. However, sometimes that the calculation is undefined for some ordered pairs, such as $a\div 0$ for any complex number $a$...
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1answer
82 views

Finding a Recurrence Relation for a binary string with n digits that do not contain 000

Consider binary strings with $n$ digits (for example, if $n=4$, some of the possible strings are 0011, 1010, 1101, etc.) Let $z_{n}$ be the number of binary strings of length n that do not contain ...
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2answers
80 views

Is there a fundamental mathematical function that requires 3 inputs or more?

So a mathematical operation can be represented as a function that maps inputs to outputs. For example "sin(x)" is a function that maps 1 input to 1 output, and "a + b" maps 2 inputs to 1 output. My ...
0
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2answers
45 views

Example of a semigroup satisfy some properties

I want to find an example a finite semigroup $S$ and $K \subseteq S$ satisfy the properties For any $a,b \in K$, we have $a,b \in \langle c \rangle$ for some $c \in S$. $K$ does not hold closure ...
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2answers
104 views

How many associative binary operations are there on a 2 element set?

We can easily find commutative binary operations on a 2 element set from the truth table (if ab=ba then the operation is commutative, thus there are 8 commutative binary operations in a 2 element set)....