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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Does associativity imply closure?

Does associativity of binary operation imply closure under this operation? Sometimes definitions of semigroup, group or vector space omit axiom of closure under corresponding operations and sometimes ...
0
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0answers
29 views

Negative representation of a binary number

I saw online that if you want to convert a binary number to a negative binary number, you add 1.However, I don't understand why you do that.In a forum I saw someone explaining the following: ...
55
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7answers
3k views

Does associativity imply commutativity?

I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea: $$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ ...
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1answer
17 views

Mathematical Properties of Logical Shift

Many programming languages feature a logical shift operator, such as $(x >> n)$, where the bits of $x$ are shifted $n$ steps to the right (or left, $<<$), and the "vacated" bit positions ...
0
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0answers
22 views

Computation for binary operation

For the binary operation on $x_7$ on set $\{1,2,3,4,5,6\}$, compute $3^{-1} x_7 4 $ Could some tell me what is $x_7$ here so that I may be able to solve this problem?
0
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1answer
30 views

Does the following function define a distance metric?

For real numeric vectors of length $N$, let $a_n \succ b_n$ be one if true and zero if false. The distance between $A$ and $B$ is $$\sum_1^N a_n \succ b_n$$ Note that this is very similar to the ...
0
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1answer
39 views

Question on the Notation of an Abstract Algebra Question

The following is a question that I came across in a textbook I'm reviewing for self-study. The book is "Introduction to Abstract Algebra", 4th Edition, by W. Keith Nicholson. I have a question both ...
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1answer
48 views

Product of sets as complexes

What does it mean to take the product of two sets of complex numbers as complexes? Reading this paper: "The Determinant of the Sum of Two Normal Matrices with Prescribed Eigenvalues" by N. Bebiano ...
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0answers
28 views

Definition of Operation

What are operations in Mathematics? I do not find it formally defined anywhere. What is the difference between operation and function? Earlier I thoght operations are just binary operations. But later ...
2
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1answer
33 views

Is every such family induced by an associative operation?

Suppose we're given an associative operation $\star : X \times X \rightarrow X$. Then for each $n \in \mathbb{N}_{>0}$, there's a function $f_n : X^n \rightarrow X$ given as follows: $$f_n(x_0,\...
0
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3answers
54 views

Can $0$ be added to any equation without changing the outcome?

I was thinking about adding $0$ to an equation, e.g.: A very simple one: $$2x + 2 = 10\\ 2x = 8 \\ x = 4 .$$ If you add "$+ 0$" to any side it does not change the outcome. $2x + 2 + 0 = 10 \...
8
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2answers
56 views

Is there a name for the operation $f^{-1}(f(x) \oplus f(y))$?

This question is inspired by and/or a generalization of this question about the "reciprocal addition" operation. Consider the following: One is tempted to say multiplication is simply "addition ...
0
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1answer
22 views

Give an example of a set with two binary operations, addition and multiplication, in which we have left distributivity but not right distributivity

Give an example of a set with two binary operations, addition and multiplication, in which the left distributive law holds but the right distributive law does not hold. I.e.: $$a(b+c)=ab+ac\text{, ...
2
votes
1answer
59 views

how to prove $G$ is an abelian group under $*$ (called the real numbers mod 1)

Let $G = \{x \in \mathbb{R}~|~0\leq x < 1\}$ and for $x,y \in G$ let $x*y$ be the fractional part of $x+y$ i.e $x*y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$. ...
5
votes
4answers
84 views

Showing associativity of (x*y) = (xy)/(x+y+1)

In order to show something is associative one must show that $(x*y)*z$ = $x*(y*z)$. I want to show that $x * y = \frac{xy}{x+y+1}$ is associative. This is for self-study (I'm learning algebra over ...
3
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2answers
70 views

Show that $G$ is a group under $*$

Let $G$ be the set of rational numbers $x$ with $x \neq\frac{-1}{2}.$ For $x, y ∈ G$ define $$x ∗ y = 2xy + x + y.$$ Show that $G$ is a group under $*$. I know how to show that associativity holds ...
4
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0answers
58 views

Does this operation have a name?

For a field $F$, define the binary operation $\parallel :(F\mathbb{P}^1 \times F\mathbb{P}^1 \setminus\{(0,0)\}) \to F\mathbb{P}^1$ by $$a \parallel b = \frac{1}{\frac{1}{a} + \frac{1}{b}}.$$ This ...
1
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1answer
28 views

What is the approach to understand this algorithm?

Given $\{x_1, x_2,\ldots x_n\}$ where $x_i \in \{0, 1\}$ there is a binary equation $\varphi$ that is $x_{t_1}+x_{t_2}+\cdots+ x_{t_m}=0 \mod 2$ where $t_i \in \{1,2,\ldots,n\}$ for $x≥1$, $i=1,2,\...
1
vote
2answers
14 views

Can the state of a system after applying the operation “absolute value” be got back using elementary operations or transformations?

Take the operation or transformation "addition". You can get back the original state of the system by doing the opposite operation, i.e., "subtraction". But, if the operation is "absolute value", you ...
0
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2answers
31 views

solving an XOR matrix

I'm working on a somewhat-unique linear algebra problem arising from XORing files together in order to encode them, and then figuring out how to subsequently recreate the original files from the ...
0
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1answer
18 views

Binary Decision Trees

I know the basics to a binary decision tree, but this problem has me a little stumped, and I'm looking for some verification on my ideas. The problem is: "Create a binary decision tree that reflects ...
1
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1answer
20 views

Anticommutative binary operation not commutative

A binary operation $\circledast$ on a set $X$ is called anticommutative if $\exists r\in X: x\circledast r = x,\;\; x\in X$ and $x\circledast y=r\Leftrightarrow (x\circledast y)\circledast(y\...
0
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1answer
26 views

How we can represent $a^b$ in following form

Consider $$a^b= a ^ {101101} $$ As if we split the binary representation of $b$, $$b = 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2+ 0 \cdot 2^1 + 1 \cdot 2^0 $$ Then how are we able to ...
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3answers
56 views

Why 1/1010 is 0.0001100110011001 [closed]

Can someone please demonstrate why 1/1010 in binary is 0.0001100110011001...? I've tried doing the math and I don't get the same result. Thanks in advance!
0
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0answers
15 views

Binary operations being continuous under a topology?

For a set $S$ and some function $f: S \rightarrow S$ and $a\in S$, $f$ is continuous at $a$ under a topology $N$ if for all neighborhoods $N_1(f(a))$ there exists a neighborhood $N_2(a)$ such that $f(...
0
votes
3answers
36 views

Proving that Z with the binary operation is a monoid?

Let $*$ denote the binary operation defined on the set $\Bbb Z$ of integers, where $$x * y = 3xy - 5x - 5y + 10$$ for all integers $x$ and $y$. Prove that $\Bbb Z$, with the binary ...
1
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1answer
19 views

Binary floating point subtraction

(In binary environment) 0.100011 * 2^6 - 0.111001 * 2^3 = 0.100011 * 2^6 - 0.000111001 * 2^6 = 0.100011000 * 2^6 + 1.111000111 * 2^6 (convert left part into 2's complement) = 10.011011111 * 2^6 ...
0
votes
0answers
22 views

Characterization of elementary arithmetic operators to explain certain properties in programming languages

In the LISP-like family of programming languages, the four elementary arithmetic operators behave differently: + and * can take ...
0
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1answer
17 views

Induction with associative binary operation

Let * be an associative binary operation on a set 'A' with identity element e. Let 'B' be a subset of 'A' that is closed under *. Let b1, b2, b3, ... bn ∈ B. Prove that b1 * b2 * b3... bn ∈ B. ...
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1answer
20 views

Simple example of idempotent but not commutative nor associative binary operator?

Is there a simple example of a binary operation that is idempotent, but not commutative nor associative?
0
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1answer
29 views

Limits and the Distributive Property?

I am currently relearning calculus after ages of not having much to do with it, and I'm looking at the proofs for the basic limit laws. I was wondering if there is a "simpler"/ more elegant way of ...
0
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0answers
56 views

Number of Distinct Integers in Base-3

Given: An integer $A$ and a natural number $K$. XOR operation in base-3 on bits $x \ and \ y$ is defined as follows, $$ x\newcommand*\xor{\mathbin{\oplus}}y=1,⇔ x=y \\x\newcommand*\xor{\mathbin{\...
0
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0answers
122 views

Binary counting problems

Recall that counting from 1 to n in binary takes $\Theta$(n) steps; i.e., the increment operation has constant amortized cost as opposed to $\Theta$(logn) in the worst-case. a) Analyze the amortized ...
0
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1answer
23 views

Which is the correct beginning to prove that $T$ is closed under $*$

Suppose that $*$ is an associative and commutative binary operation on a set $S$. Let $$T = \{a \in S \, : \, a*a = a\}$$ Is one of these methods or both a correct approach in beginning to solve ...
0
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0answers
28 views

Division by power of 3.

Is there any fast division algorithm to divide a binary number by power of $3$. I want to find the $q,r$ for $a=q*3^b+r$, $b$ is constant.
0
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3answers
40 views

How to repeat a byte number inside another number without iterating? if possible…

Ok, I need a bit of help from my Math/Computer geeks out there. In the curse of an optimization for a program I am writing in Python, I found the following problem: for a given byte value, I need to ...
2
votes
2answers
35 views

Why in order to be a binary operation on $S$, each element of $S$ has to appear 'once and only once' in each row and column in Cayley Table?

I was reading about Composition table or Cayley Table; one of the points my book presents is that If all the entries of the table are elements of set $S$ and each element of $S$ appears once and ...
0
votes
0answers
20 views

Classes of binary operations between functions

Let $f,g : D\to \mathbb{R}$ be two functions defined from a domain $D\in \mathbb{R}$ to $\mathbb{R}$. I am looking for classes of binary operations $\circ$ between $f$ and $g$ that produce an $h:=f\...
0
votes
1answer
30 views

Decompose binary into decimal units, tens and hundreds

I have a 9 bit binary sequence (from 0 0000 0000 to 1 1111 1111) and I'd like to decompose into decimal units, tens and hundreds. Consider the following: 0 0111 1011 ==> 123 I'd like to ...
0
votes
2answers
30 views

Finding unity and units of a binary operation

Let $\ast$ be the binary operation defined on $\mathbb{N}$ by $m \ast n = \max (m, n)$; the largest of $m$ and $n$. Decide whether unity exists and if so, find the units. I know that unity is defined ...
0
votes
0answers
18 views

Encoding and decoding with bitwise XOR and Shifts

This answer may exist somewhere already but if it does I've had trouble finding it. This is based on the problem from a programming site here $$encoded\_value(x) = { x \oplus (x<<1)|x, encoded\...
1
vote
1answer
17 views

Binary operation - finding identity

Here is the question: The binary operation * is defined for $x,y \in\ S = {0,1,2,3,4,5,6}$ by $x*y=(x^3y-xy)mod7$ Find the element $e$ such that $e*y=y$ for all $y\in\ S$ So far I have the ...
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1answer
21 views

Mathematical Expression for multiplication and addition of binary arrays

Assuming I have two binar arrays with 3 bits. Array A = 1,1,0 and Array B = 0,1,1. Now I want to perform bit-wise AND operation ...
0
votes
0answers
53 views

Group Theory and Cardinals [duplicate]

I was wondering the following: Let's suppose that $G$ is a non-empty set, then, can we always find a binary operation $*$ such that $(G,*)$ is a group? For example, if we fix $G=\mathbb{Q}$ the sum ...
0
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4answers
68 views

Right way to show associativity.

Authors usually write that $*$ is associative on a set $S$ if, $(a*b)*c=a*(b*c)$ $\forall a,b,c \in S$ I think it should have been, $(a*b)*c=a*(b*c)=(a*c)*b$ $\forall a,b,c \in S$ I made all ...
2
votes
2answers
42 views

Find a fraction given the repeating binary expansions

I can do binary expansion given a fraction just fine but the question I'm stuck on says: Find fractions for the numbers with the following binary expansion: (i) $0.00\overline{110}$ and (ii) $0.0\...
29
votes
10answers
2k views

Non-associative commutative binary operation [duplicate]

Is there an example of a non-associative, commutative binary operation? What about a non-associative, commutative binary operation with identity and inverses? The only example of a non-associative ...
0
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2answers
48 views

With 8 bits is it possible to obtain an integer in more than one way? [duplicate]

This is just a curiosity that just came to my mind while thinking at IP addresses. A byte is composed of 8 bits. A bit can either be $0$ or $1$. IPv4 addresses are composed of a group of 4 bytes. ...
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3answers
109 views

Name of this property: if $x * x = y * y \implies x = y$

Algebraically speaking, what's the name of this property?: $x * x = y * y \implies x = y$ $*$ being a binary operation
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2answers
273 views

Symbol for “the greater of the two values”

I'm looking for an operator that returns the greater of two values. Here's an example. If $a=5$, $b=6$ and $???$ is the operator, I'd like to have $x$ equal $b$ when I do $x=a???b$, since $b$ is the ...