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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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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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 ...
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2answers
44 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
104 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
105 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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0answers
125 views

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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2answers
80 views

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
282 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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0answers
8 views

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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4answers
976 views

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 ...
2
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1answer
29 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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1answer
31 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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3answers
84 views

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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0answers
27 views

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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31 views

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 ...
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3answers
561 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
27 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
52 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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0answers
15 views

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 ...
4
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1answer
27 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
38 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
33 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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2answers
39 views

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 ...
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1answer
26 views

Binary subtraction with borrowing

Consider two number which we are going to subtract: 0x44 - 0x29. How can we subtract using borrowing method. The numbers in () shows the value of the digit in ...
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1answer
53 views

Can two points be added?

Can two points be added? The reason I ask is because when I think about it all I see is vector addition. I understand the difference between vectors and points. I know we used to talk about points on ...
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0answers
13 views

Binary relations for Cobb-Douglas

I am reviewing old midterms to prepare for my upcoming midterm and ran across this question: Let $\alpha , \beta \in (0,1)$. Now, let $f_{\alpha}$ and $f_{\beta}$ on $\mathbb{R^2}$ be defined as ...
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2answers
51 views

Question about some properties of the binary operation $a*b=a-b$.

Originally, the question asks whether $\mathbb{Q}^2\to \mathbb{Q}, (a,b) \mapsto a-b$ is associative, commutative, has a neutral element and has an inverse. Associativity and commutativity: In ...
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1answer
23 views

Associative or Commutative of Binary Operation

I need to figure out whether these binary operations are commutative or associative. And then whether a unity exists (but I don't know what that means). M=$\mathbb{Z}$; a*b=a-b M=$\mathbb{Q}$; ...
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1answer
46 views

Binary multiplication for negative numbers

The question is about binary multiplication for negative numbers. Assume we want to multiply -5 * -3 so the result is +15. 1) ...
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0answers
20 views

Binary operations on Sets and Maps

Hi i'm just attempting a question on sets and maps but i'm a little confused by my lecturers notation and was hoping someone could help. Ive got the sets X, Y and Z. π is the mapping of X to Y and ψ ...
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49 views

Checking arithmetics

Is there any known way to check the correctness of a binary arithmetic operation (esp adding) without actually doing the operation?
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0answers
20 views

Binary operations and proofs

$\text{binary}(a)$ = binary representation of a base 10 number $a$ Are the following statements correct? If yes, where can I find the proofs? (1) $\text{binary}(a\times b)=\text{binary}(a)\times ...
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1answer
26 views

Properties of x*y = |x+y| in set R

This is the second exercise in the second set of chapter 2 in Pinter's A Book of Abstract Algebra. It is asked to check for commutativity, associativity, identity element in $\mathbb{R}$ and inverse ...
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1answer
32 views

Multiplication Table for $\Bbb Z_9^*$ [closed]

Define $$ \Bbb Z_n^* = \{ x:x \in \Bbb Z_n \text{ and } x \text{ is relatively prime to } n\} $$ If $G = \Bbb Z_n^*$, then the binary operation is multiplication modulo $n$. How to construct the ...
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3answers
125 views

Sets on which composition of bijective functions is commutative.

I have the following problem: Let $X$ be a set and $F$ the set of all bijections $f: X\to X$. We consider the composition $f \circ g$ as a binary operation on $F$. Describe precisely for which sets ...
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1answer
27 views

Nullary and unary operations defined on a uniquely complemented lattice?

A lattice is a set $L$ with two binary operations, $\lor$ "join" and $\land$ "meet". In a complemented lattice, for every element $a$ there exists an element $a^{\perp}$ such that $a \lor a^\perp=1$ ...
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1answer
54 views

Proving commutativity of a binary operation?

While it's easy to show that something isn't commutative, it's not so easy to prove that something is anything – at least for me! If we take a relatively simple example: $(\mathbb{Z}_{n}$ , ...
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1answer
32 views

Proving two elements of a set are equal based on a two-sided identity

Say I have a set S w/ an associative binary operation *: S x S -> S and a two-sided identity e, and let . Let L and R be elements of S such that L * s = e = s * R How can I prove that L = R ? Since ...
1
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1answer
31 views

The number of distinct partial binary operations on a finite set of n elements

I am asked to show that there are exactly $(n+1)^{n^2}$ partial binary operations on a finite set of n elements. My professor said that this can be done using a combinatoric argument, but I have ...
0
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1answer
91 views

How do you prove the left and right side of an identity of a set are equal?

I'm having some trouble understanding sets w/ associative binary operations. Say I have a set "S" w/ the associative binary operation SxS -> S. If 'L' is a left identity of S and 'R' is a right ...
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2answers
55 views

Suppose that $\cdot$ is associative and has an identity element. Show that an element $g \in G$ has at most one inverse

Let $(G,\cdot)$ be a group with $e$ its neutral element. For an element $g\in G$, there exists one inverse element in $G$, denoted by $g^{−1}$, such that $g\cdot g^{−1}=g^{−1}\cdot g=e$. Can this be ...
4
votes
1answer
134 views

Why calculating XOR of consecutive values can be simplified? [duplicate]

I was trying to calculate integer xor of 0..n. I named the function xored(n). Note that in examples below ^ does not mean power but integer xor (like in C or Java language) So, xored(0) = 0, ...
3
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1answer
46 views

How to solve (0.1 - 10.3 + 5.132)/12.8 and round off correctly?

I've recently learnt the rules about rounding off when adding/subtracting and when multiplying/dividing. I know that when you add/subtract, the number of decimal places in the result should equal the ...
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1answer
46 views

Binary word addition; error pattern

If a word $a = a_1,a_2,...,a_n$ is sent is sent (this is in regards to coding/IT/etc.-I'm trying not to include any extraneous information) and a word $b= b_1,b_2,...,b_n$ is received (where the ...
0
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1answer
34 views

Identity element of word addition

I realize this is rather an arbitrary question, but it's important to me, that I understand it and get it right, and I'm not finding the answer anywhere else. I'm working through "A Book of Abstract ...
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0answers
61 views

KoreK's ChopChop Attack (Inverse Arbaugh Inductive Attack)

I would like to know how KoreK's "ChopChop" attack on the WEP protocol works on a basic mathematical level. His original source code provides little clue as it uses precomputed values, the origins of ...
3
votes
0answers
122 views

Is the operation $x*y=xy/(x+y+1)$ associative or not?

This questions is from Charles C. Pinter's "A Book Of Abstract Algebra, Second Edition". This is the 7th question of Chapter 2 on page 24. In the selected answers section it says it is not associative ...
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1answer
58 views

Operation table for A+B where + denotes the operation of symmetric difference

If someone could please verify if my operation table in the picture below is correct it'd be much appreciated. The task was given: $P_D=\{A: A \subset D\}$ and $D$ is a $3$-element set $D=\{a, ...
3
votes
1answer
39 views

Convolution can smooth an input function, is there an operation which bunches it up?

An easy to remember description of what the convolution of two functions is, is to say that one is a weight function and the result is a weighted average of the other function. The canonical example ...