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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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 ...
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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 ...
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1answer
18 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 ...
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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
47 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!
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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 ...
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33 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 ...
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1answer
18 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 ...
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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 ...
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16 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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16 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?
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25 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 ...
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50 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 ...
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0answers
119 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 ...
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1answer
19 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 ...
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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.
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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 ...
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2answers
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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 ...
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18 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 ...
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1answer
27 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 ...
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2answers
24 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 ...
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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, ...
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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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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 ...
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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 ...
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66 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 ...
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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) ...
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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 ...
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2answers
47 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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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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171 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 ...
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Properties and geometrical interpretations of a specific planar vector law.

Genesis of the following thoughts. One usually define the following internal composition law on $\mathbb{R}^2$: $$+:\left\{\begin{array}{ccc} ...
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efficient way to express large numbers

I recently watched the walkthrough of Graham's Number on YouTube (Numberphile). Mind-blowing of course. I then puttered around in other large number topics like Ackerman and Tree(3) and fast growing ...
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1answer
287 views

Are the addition and multiplication of real numbers, as we know them, unique?

After recently concluding my Real Analysis course in this semester I got the following question bugging me: Is the canonical operation of addition on real numbers unique? Otherwise: Can we define ...
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How would I round the number $-(1.01101|01)_2$ down?

I'm trying to round $-(1.01101|01)_2$ in binary down. The $|01$ are the guard bits. The answer is $-(1.01110)_2$ My attempt is as follows: $-(1.01101|01)_2$ $-(0.00001|00)$ = $(1.01100)$ Not ...
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A name for this property?

Let $*$ be an operation such that $(xy)^* = y^*x^*$, e.g. if $x,y$ are $2\times2$ matrices and $*$ is "take the inverse" or if $x,y$ are operators and if $*$ is the adjoint. Is there a name for such ...
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1answer
34 views

Prove the following set endowed with the binary operation is an abelian group

Let $∗$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x ∗ z = y$. Show that this set with the operation ...
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33 views

What actually determines a 'borrow' in binary subtraction?

Given Byte1 and Byte2 in binary. How to determine whether Byte1 - Byte2 results in a borrow ...
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So there's addition, multiplication, exponentiation and tetration, but is there a level before addition? [duplicate]

I've tried to do this with a sheet of paper since I knew about this: $$x+x = 2x$$ $$x\cdot x = x^2$$ $$x^x = {^{2}x} $$ So based on logic, I assume that: $$x.x=x+2$$ The dot in between the two xs ...
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is there no needs three or more binary operations for constructing(?)algebra?

In Algebra class, professor metioned godel proved no three or more operation be needed in algebra. if he is right where can i get the proof. if you can't get it, Sorry for my short english..
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How to find the number of subsets with a given length and XOR?

I have A (0<A<500000) elements (up to 10^6) in the set. I need to find in how many ways can I remove a subset, the size of ...
10
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3answers
564 views

Why do empty operations give the identity?

Why is it that empty operations give the identity of that operation? Wikipedia says that the empty product is taken to be 1 "by convention", but this convention is consistent with ideas that seem ...
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2answers
28 views

Equivalence relations and binary operations

Let S be the set of all sequences of real numbers. Define a relation $\sim$ on S by $\{x_n\} \sim \{y_n\}$ if $x_n - y_n \rightarrow 0$. (i) Prove that $\sim$ is an equivalence relation. (ii) Let ...
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1answer
53 views

Associative operation on a set $S$

Let $S$ be a non-empty set and $n\in \mathbb{N}, n\geq 2$ a fixed integer. We consider an associative operation $"\cdot"$ on $S$ with the following properties: $x^{n+1}=x, \forall x\in S$ ...
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binary divison with remainder

I wanted to calculate this (binary): 10101.101/1.1. the result is: 1110.01101010101010101010101010... I succeed in calculating the integer part, but I didn't understand how I find the numbers that ...
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1answer
35 views

Prove the commutativity of addition from the existence of left and right additive inverse.

Recently I have encountered an exercise from the book A survey of modern algebra by Birkhoff and MacLane, which states: Let R be a set equipped with two operations (addition and multiplication) that ...
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1answer
47 views

Transitivity, symmetry on empty set X, non-empty relation R [closed]

If I had an empty set X, with a relation R containing elements 1 and 2 In my directed graph if I had (1,2) and (2,1), would I still have transitivity and symmetry even though this is an invalid ...
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1answer
13 views

binary relations defining an equivalence relation on S

Is this a true statement for binary relations defines an equivalence relation on S: S is the set of all n-digit binary sequences. We say that two binary sequences are in a relation if and only if ...
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1answer
46 views

Binary operations 1011 & (~0 << 2)

My thought process to solving this is that 1011 & (~0 << 2) = 1011 & (1 << 2) = 1011 & 0100 = 0000. But my book says the answer is 1000, ...
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Is it possible to have a symmetric and transitive relation on $\mathbb Z$ that isn't reflexive?

I was wondering this the other day and I have been trying to come up with an example for a bit but can't produce one. If there is a proof that such a thing couldn't exist I would be interested in ...