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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34 views

Counting binary operations on a group [closed]

For a finite set S with n elements, how many binary operations can be there which makes S a group?
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Proof for the bitwise XOR of two numbers to have Odd Number of set bits only when exactly one of them have odd number of bits?

I was doing some Algorithms Contest Question ( OffCourse it is not live now), Where I encountered this amazing Pattern that Two Non Negative Numbers $A$ and $B$ whose Exclusive XOR represented by $C=...
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Factoring a XOR Expression in an Equation

I have an equation like this: $(i_1 \cdot x) \oplus (i_2 \cdot x) = r$ where $i_1, i_2,$ and $r$ are known values and $\oplus$ is xor. What I need is that same equation but expressed as $i_3 \cdot x = ...
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1answer
56 views

Can every group be embedded in a non-associative group? [closed]

A non-empty set S that satisfies all group axioms except associativity is called a non-associative group. If $G$ is any group then is it possible to embed $G$ in some non-associative group?
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Number of steps needed to show that a binary operation is associative on n operand

I was trying to prove a binary operation is associative on a given number of the operand. I did it, but then I checked the textbook. I took 5 lines, but the book proved it in 6 lines using a precise ...
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1answer
22 views

Are Linear Complexes and Simple Graphs the same thing?

I refer to the background of the notion of $Linear~complex$ by A. A. Zykov in the article entitled "General properties of Linear Complexes" A $Linear~complex $ (or simplex complex) is a ...
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1answer
17 views

Associativity of binary operations on a two-element underlying set (is there a pattern?)

The overall problem is to establish, which binary operations on a two-element set $A=\{a, b\}$ are commutative and associative. There are 16 of them altogether, obviously, analogous to operations on ...
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2answers
21 views

For $G_1$, $G_2$, $G_3$ simple undirected graphs on the same vertices with disjoint edge sets, if $G_1\cup(G_2\cap G_3)=G_2$, then $G_1=G_2$

If $G_1, G_2$ and $G_3$ are simple undirected graphs on the same set of vertices with disjoint edge sets. If we have a graph equation $$G_1\cup(G_2\cap G_3)=G_2$$ Then we have to show that $G_1=G_2,$ ...
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20 views

How can I prove that an invented operation works with negative numbers? [duplicate]

Just out of curiosity I tried making a new operation to see its properties. It is like a variation of addition that works like this: a ¨ b = a + b - 1 For example, I found that it has the commutative ...
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1answer
22 views

distributive and non-commutative binary operation on real numbers

I've been searching for a binary operation that is distributive over addition but not commutative on $\mathbb{R}$; to be more exact I want to have $$a(b+c)=ab+ac$$ $$(a+b)c=ac+bc$$ for all $a,b,c \in \...
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1answer
23 views

The set of all nilpotent elements of a weird binary operation.

On the set of integers $Z$ define the binary operations: Addition: a#b=a+b+1; Multiplication :a*b=a+b+ab; For a, b in $Z$. $Z$ is a ring. With the zero element= $-1$ and a unity= $0$. Find the set of ...
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Question regarding finding SOP using K Map derived from Truth Tables.

So, I'm new to logic design, and I was given this question about a circuit that takes a 5-bit input ABCDE which is a value in excess 8 representation. An output K is produced by the circuit which is ...
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29 views

Understanding on Artin's proof of the generalised associative law for associative binary operation

On the 2nd edition of Artin's Algebra, the writer uses proposition 2.1.4 to imply that generalised associative law works for associative binary operation: Proposition 2.1.4 Let an associative law of ...
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1answer
71 views

Finding the result of $1/2*1/3…*1/1000$

Let the binary operation $x*y=(x+y)/(1+xy)$ where $x,y\in (-1,1)$ Find $(1/2)*(1/3)*...*(1/1000)$ I found that $1/x*1/y=(x+y)/(1+xy)$ but I can not find a rule for $n$ such numbers and apply induction ...
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Binary addition using Linear algebra

Is it mathematically possible to do binary addition using linear algebra? To be more precise, when the binary numbers are represented by vectors, each element containing respectively the ones, tens, ...
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21 views

Internal binary operation

Dirichlet Convolution. If $f,g:\mathbb {N} \to \mathbb {C}$ are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution $f ∗ g$ is a new arithmetic ...
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32 views

Name for operation with property x # x = x

Like, for example, min has this property because $min(x,x) = x$ Also, is there another, more specific name if this operation is also associative and commutative
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1answer
67 views

Convert Hexadecimal $8D(16)$ to binary in signed magnitude

I'm supposed to covert hexadecimal value, $8D(16)$ into $8$-bit binary if signed magnitude representation is used. $8D(16)$ $\to$ $1000$ $1101(2)$ For signed magnitude, the left most bit is used to ...
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A direct formula for counting 1s in binary form of a positive integer

The question is edited based on the @Damien comment. Question: Is there a direct formula (not recursive) for counting 1s in binary form of a positive integer? For example, suppose the formula is ...
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1answer
21 views

How to show nonassociativity of the positive rationals under a binary operation defined in terms of max and min?

Consider $\mathbb{Q}^+$ with the usual $\leq$ relation and the binary operation $\circ$ defined as: $$p \circ q = max(p,q) + \frac{1}{2} min(p,q)$$ A book that I'm reading states that the operation $\...
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3answers
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Does the set intersection operation has unit

I'm trying to get better understanding of binary operations, and I came across this problem: namely on one online discussions I saw that set intersection as binary operation doesn't have a unit, ...
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Are all commutative, associative binary operations isomorphic to addition?

Addition and multiplication are the two classic commutative, associative binary operations on the reals. They satisfy a striking property: they are equivalent up to unary operations. By taking a ...
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Operations on functions with different co-domains

How are operations on functions with different co-domains defined? Let f: D1 -> C1 and g: D2 -> C2, and let (f + g): D -> C What are sets D and C in terms of D1, D2, C1, C2? Furthermore, is ...
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1answer
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Are these variations of complex multiplication studied topics?

Complex multiplication is very well understood geometrically and algebraically, but I wonder what about the following operators -angles assumed to be randians $[0,2\pi)$: Complex multiplication(...
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1answer
35 views

finding a four elements group and binary operation upon it that has closure, associativity, an identity element and an inverse to each element.

the question is in the heading. I tried using the group {0,1,2,3} and the binary operation $*$ defined as $a*b=|a-b|$ which I proved to have closure, an identity and an inverse, but am unable to prove ...
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Confusion about binary expansion of number in $(0,1)$.

Unfamiliar with discrete mathematics, I wondered in the following two answers Fast way to find period-n points of a tent map? Techniques for finding period points , why the authors wrote: 1. "...
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1answer
49 views

Let $(A,+, \cdot)$ be a ring with the given tables, determine which is the zero and inverse of each element

Let $(A,+, \cdot)$ be a ring with the given tables for operations $+$ and $\cdot$ \begin{array}{c|cccc} + & s & t & x & y \\ \hline s & y & x & s & t \\ t & x &...
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1answer
74 views

Given a set A in which all elements are of the form: $x+y \sqrt3$; $x,y \in \mathbb{Q}$. What structure can you define…

Given a set $A$ in which all elements are of the form: $x+y \sqrt3$, $x,y \in \mathbb{Q}$. What algebraic structure can you define with operations of addition and multiplication? I am stuck figuring ...
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2answers
63 views

Does distributivity implies commutativity of one operation

Suppose there is a set $S$, equipped with two binary operations, $*$ and $@$, such that S is closed and associative under both the operations. There exist inverses and identity with respect to both ...
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Why 1 isn’t $a*b=max(a,b)$ identity element?

I had an exam this morning and the first question was in a test form ,the question goes: Imagine there is a binary operation $a*b=max(a,b)$ on N . Which option is correct ? a)$(N,*)$ is not abelian. ...
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52 views

Find path through binary tree to get to desired node #

Say we have a binary tree. If know how many nodes X there are in the tree, how can we navigate from the root node to the node with value X without any backtracking? I am doing binary arithmetic and ...
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1answer
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Given, (F,+,•) and (F,+,×) are fields; then • and × are same binary operations

(While reading about classification of simple groups, I was wondering if there could be something like simple fields or something... And this thought came in mind) When we talk about field in abstract ...
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Addition vs multiplication as binary operators

It is known that arithmetic operations of addition and multiplication are binary operations that take in two inputs and give out a single output. However, consider the following scenario: Suppose I ...
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1answer
26 views

Arithmetic operation

Given $$9 ∆ 4=20,$$ $$7 ∆ 3=12,$$ $$10 ∆ 8=16,$$ find $m ∆ n$. I have tried so many ways to get this but I may be missing something.
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1answer
121 views

Is there a deeper reason why exponentiation is not associative?

Addition can be thought of as repeated counting; multiplication can be thought of as repeated addition; and exponentiation can be thought of as repeated multiplication. And yet, while the first three ...
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1answer
44 views

For which type of input does this operation repeat the outcome eventually and run forever?

This problem is a part of another (programming) problem on which I am working, the start of which requires us to identify a certain pattern, I believe. We are given a pair of positive integers x and y....
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1answer
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Binary operation or Binary relation? [closed]

$Y$ corresponds to set of all the binary relations over set $X$. If $R$ is the composition of binary relation on set $X$, is $R$ a binary operation or a binary relation?
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28 views

Bijectivity of a binary function

How to prove that any binary function (Boolean function) is bijective in $x$ if and only if the function is linear in $x$. For example, $f(x_{1}, x_{2}, ..., x_{n}) = x_{1} + x_{2}*x_{3}*...*x_{n}$ is ...
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25 views

How does all 1 exponent and all 0 fraction represent infinity?

I am studying IEEE 754 FLOATING POINT STANDARD. Standard says The number is infinity when: e (Biased Exponent) = 255 f = 0 I am unable to understand this because if fraction (f) = 0 and exponent = ...
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32 views

Is it possible to define some form of relationship between two operations?

I am thinking we have transformations and all forms of relationships on sets. But thinking for an abstract algebraic theoretical session, I pondered on whether its possible to transform (or just ...
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167 views

If not associative, then what?

Consider a binary operation $*$ acting from a set $X$ to itself. It's useful and standard to work with operations which are associative, such that $(a*b)*c = a*(b*c)$. What about operations which are ...
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41 views

Does this graph join like operation have a known name? Any special algebraic property?

I am defining a graph operation $\pmb\nabla$ in the following. The notation $\pmb\nabla$ is a modified form of graph join operation, and defined as follows. Consider two graphs $G$ and $H$, then $G\...
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1answer
42 views

Prove that certain operation is commutative [duplicate]

Suppose we have a set $X$ and a binary operation $\circ : X \times X \to X$, such that $\forall x,y \in X$ the following equalities hold $$y \circ (y\circ x) = x, (x\circ y)\circ y = x.$$ How can ...
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1answer
53 views

Is the graph of the following type possible?

Let a simple undirected graph be denoted by $(V, E)$, where $V$ is the set of vertices and $E$ is the set of edges. Let each vertex of a graph be a natural number. Now, I want to define the graph's ...
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30 views

Question on binary operations

A question defines A = {1,2,3,4,5,6} and a binary operation * such that $a*b=r$, where r is the least non-negative remainder when the product $ab$ is divided by $k$. Find k for * to be a binary ...
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1answer
40 views

If R,S, and T are binary relations on a nonempty set, which of these following statements are true and why? [closed]

$T \circ (R \cap S) = (T \circ R) \cap (T \circ S)$ $T \circ (R - S) = (T \circ R) - (T \circ S)$ $(R \cup S) \circ T = (R \circ T)\cup(S \circ T)$ $(R-S)\circ T = (R \circ T ) - (S \circ T)$
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1answer
44 views

My wrong proof about $\mathcal R\left[\bigcap_{i \in I} C_i\right]=\bigcap_{i \in I}\mathcal R \left[C_i\right]$

Let $\mathcal R \subseteq A \times B $ be a binary relation and $\left\{C_i\right\}_{i \in I} \subseteq A$ is a family of subsets index by $I$,then :$$\mathcal R\left[\bigcap_{i \in I} C_i\right]\...
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1answer
179 views

Determine whether or not the given $*$ is a binary operation on the given set S.

$S = \mathbb{Z}, a * b = a+b^2$ Commutative: $a*b = b*a$ $a*b = a + b^2$ and $b*a = b+a^2$ and they aren't the same at all. Associative: $(a*b)*c = a*(b*c)$ $(a*b)*c = (a+b^2)* c = a+b^2+c^2$ and $a*(...
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1answer
50 views

Fill in a partly filled in table such that it makes the magma $(M,*)$ associative, commutative, has an identity element and has no zero-elements.

Below is a partly filled in table for a binary operation ($*$) on the set $M=\{a,b,c,d\}$. I am trying to fill in the rest such that the magma $(M,*)$ becomes associative, commutative, has an identity ...
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2answers
56 views

Prove that the groups $(\mathbb{Z_n}, +)$ of residue classes modulo $n$ and $(U_\mathbb{n}, \cdot)$ of the $n$-th roots of unity are isomorphic.

I have to prove that the groups $(\mathbb{Z_n}, +)$ and $(U_\mathbb{n}, \cdot)$ are isomorphic, where $\mathbb{Z}_n$ is the set of residue classes modulo $n$: $$\mathbb{Z_n} = \{\hat{0}, \hat{1}, ..., ...

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