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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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Conservative idempotent magma - proof attempt

I need help with checking proof about idempotent and conservative magmas. Let magma be any ordered pair $(M, \odot)$, where $M$ is nonempty set and $\odot$ binary operation on $M$. Now I need to ...
Oliver Bukovianský's user avatar
-1 votes
1 answer
56 views

View minus sign as operator or part of the number? How to differentiate?

I came across this problem,looking at the distributive law "a*(b+c) = ab+ac" / "a*(b-c) = ab-ac". Lets say we have the following term: -4 * (2 - 4) What would you say is c? Is c -4 ...
derflo's user avatar
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8 votes
3 answers
1k views

How to deal with multiple plus-or-minus signs (±) in a single expression

When you have a single plus-or-minus symbol, the meaning is clear: $a±b = (a+b) OR (a-b)$ When you have plus-or-minus and minus-or-plus symbols, the meaning is also clear, as described in many places, ...
Adamimoka's user avatar
2 votes
2 answers
156 views

priority of operations in function composition is backwards

I feel like the priority of function composition is backwards, and I would like to have a deep understanding of the phenomenon. I do understand that function composition reads right to left: $$(f\circ ...
Victor Daniel's user avatar
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0 answers
53 views

Ranking and unranking of a binary subset

Let's consider "N" bits. We want to rank and unrank a specific subset of bit combinations based on the following criteria - ...
Dave's user avatar
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1 vote
0 answers
32 views

Could we define a fifth arithmetic operation on real (or complex) numbers that is independent of addition, subtraction, multiplication, and division?

The four basic arithmetic operations with real (or complex) numbers are addition, subtraction, multiplication, and division. the first two being inverse operations and the last two being inverses of ...
Saaqib Mahmood's user avatar
0 votes
1 answer
45 views

Classification of binary associative operations on $\mathbb{R}^n$

Is there an explicit characterization of associative binary operations on two vectors of the same dimension? Some examples include component wise +/*/max, or matrix multiplication if the vectors can ...
John Jiang's user avatar
1 vote
0 answers
53 views

Problem about binary/decimal bitwise and operation

I found a (maybe) fun problem, but I could not solve this. $10110_{(10)} \& 10011_{(10)} = 10010_{(10)}$ $10110_{(2)} \& 10011_{(2)} = 10010_{(2)}$ $\&$ is the bitwise and operator. Their ...
Vermeil's user avatar
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1 vote
1 answer
58 views

Problem 27. Integrate a conditional BITWISE right-shift function.

This problem has been haunting me since I've seen it on Facebook. The function $f$ converts $1$s into $01$s when written in binary, e.g., when $a = 0.101$_bin, $f(a) = 0.01001$_bin. Compute the ...
Matthew Jones's user avatar
3 votes
6 answers
573 views

The operation $ (a,b)(c,d)=(ac-bd,ad+bc) $ on $\Bbb R\times\Bbb R\backslash (0,0)$ yields a group

Here is the binary operation $ *: \mathbb{R}\times \mathbb{R} \backslash (0,0) $ defined by $ (a,b)(c,d)=(ac-bd,ad+bc) $. My idea is that to show this is a group ($\mathbb{R}\times \mathbb{R} \...
Jackanap3s's user avatar
2 votes
2 answers
142 views

Why do k arithmetic right/left bit shifts divide by $2^k$/multiply by $2^k$ in two's complement, rigorously?

I want to understand the semantics of rights bit shifts x>>k in two's complement properly, in particular why do right bit shifts of size $k$ approximately ...
Charlie Parker's user avatar
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0 answers
31 views

Which of the following is a binary operation?

This question was asked to me by a student of mine and I am not able to make any progress on this question. Let $A=${1,2,3,4,5} , then which of the following is a binary operation? (a) {((1,1),1),((...
user avatar
0 votes
1 answer
67 views

What do the symbols for these binary operations on a set mean?

If S = {0,1,2,3,4} and (a,b) is an arbitrary ordered pair such that a ∈ S and b ∈ S, which of the matchings in Exercises 12-15 are binary operations on S? Construct operation tables. (a,b) ----> a ...
Anthony Baldini's user avatar
1 vote
2 answers
52 views

Is a commutative and associative with neutral element operation and inverses on R^2 necessarily componentwise sum?

I'm trying to make a derivation of the standard operations of complex numbers from field axioms and the condition that operations on real numbers work the same way. One part of that work is proving ...
Andrea Miele's user avatar
2 votes
1 answer
72 views

Binary operator and their inverses.

Having trouble finding it, but there was a question posted last week regarding a binary operation $x,y \in \mathbb R, x\oplus y=\frac{xy}{x+y}$. It also addressed Resistors in parallel and worked ...
TurlocTheRed's user avatar
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Operation that is distributive over convolution of integer sequences

I'm working in the spaces $\mathbb{I}_Z$ and $\mathbb{I}_Q$ of infinite integer/rational sequences that start with $1$: $$∀x∈\mathbb{I}_Z\ ∀n∈\mathbb{N}:x_0=1,x_n∈\mathbb{Z}$$ $$∀y∈\mathbb{I}_Q\ ∀n∈\...
Aberone's user avatar
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3 votes
0 answers
79 views

When can a partial associative operation be extended?

Let $X$ be a set with a partial operation $\cdot$ which is associative in the sense that if $x, y, z \in X$ and $x \cdot y$ and $y \cdot z$ are both defined, then $(x \cdot y) \cdot z$ and $x \cdot (y ...
I Eat Groups's user avatar
0 votes
1 answer
76 views

Distributive property of sum and intersection of ideals in a polynomial ring [closed]

Let $S=k[x_1,\dots,x_n]$, where $k$ is a field, and $I,J,K$ be three ideals of $S$. Are the following right or are they wrong? $ I+ (J\cap K)=(I+J)\cap (I+ K)$ $ I\cap (J+ K)=(I\cap J)+ (I\cap K)$
Hola's user avatar
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5 votes
2 answers
966 views

Why can't logarithms be expressed in terms of exponentiation?

The inverse operation of addition is subtraction, and vice versa. They undo each other. Funnily enough, subtraction can be expressed in terms of addition. $a-b=a+(-b)$. Additionaly, the inverse ...
The_Animator's user avatar
1 vote
1 answer
41 views

Proving that, a particular binary operation is the only possible operation.

I attempted the following question: Show that the number of binary operations on {1,2} having 1 as identity and having 2 as the inverse of 2 is exactly one. My approach to solve the problem was this: ...
Nikhil Kumar's user avatar
4 votes
2 answers
510 views

Abstract formulation of associativity

Say we are given a binary operation $f$ on a set $X$, that is, $$ f : X \times X \to X. $$ Denote by $\text{Id}$ the identity map on $X$. We say that $f$ is associative if, for all $x, y, z \in X$, we ...
markusas's user avatar
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4 votes
10 answers
492 views

Some easy examples of operations that don't work like you'd expect them to, i.e. not being commutative or associative etc.

I'm currently teaching some undergrads in their first / second semester about fields, groups, rings and vector spaces. I often see them argue that something is definitely a field, since addition / ...
Zedssad's user avatar
  • 718
0 votes
1 answer
20 views

Maximal Extension Chain of Halfgroupoids

A book I am reading gives the following definitions: A collection $\{L_i:i=0,1,2,...\}$ of halfgroupoids $L_i$ is called an extension chain if $L_{i+1}$ is an extension of $L_i$ for each $i$. If $G$ ...
shea's user avatar
  • 31
0 votes
0 answers
34 views

Closest Equivalent to Cayley Graphs for Partial Groupoids?

[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.] This question may be nonsensical, given that the duality ...
shea's user avatar
  • 31
7 votes
4 answers
222 views

Associative binary operation on the positive reals

Let $*$ be an associative binary law on the interval $(0,\infty)$ of the positive reals. Suppose that for every $a,b,c \in (0, \infty)$ we know $$ a * b * c = \frac{abc}{ab+bc+ca}.$$ Can we show that ...
mgeorge23's user avatar
29 votes
4 answers
5k views

Can you multiply 3 matrices simultaneouly?

I know that the algorithm for multiplying 2 matrices is defined as: $$(AB)_{ij} = (\text{row }i\text{ of matrix }A) ⋅ (\text{column }j\text{ of matrix }B)$$ And I know that matrix multiplication is ...
Alice 's user avatar
  • 327
1 vote
1 answer
139 views

Rule to calculate derivative of function which is a binary operation between two other functions.

Let $\mathbf{g}:\mathbb R\rightarrow \mathbb R$ be a differentiable function defined as $g(x)=f(x)*h(x)$ where $*$ is any binary operation between the two differentiable functions $\mathbf{h}:\mathbb ...
Samar Sidhu's user avatar
0 votes
1 answer
80 views

Clarification on Multiplication in $GF(2^3)$ vs. Boolean Algebra

While experimenting with finite fields, specifically $GF(2^3)$, I stumbled upon a puzzling situation when comparing multiplication operations to those in Boolean algebra. Let's take two elements $A$ ...
ZenithZero's user avatar
1 vote
0 answers
51 views

Confusion in the definition of Algebraic structure, system, operation, magma

During studying the text book of abstract algebra by john Farleigh, I encounter with tye definition of binary Algebraic structure. Then I tried to find the difference between binary Algebraic ...
Afzal's user avatar
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0 votes
1 answer
53 views

The identity element of $G$ under $*$

Let $G$ be a set of ordered pairs $(a, b), \, a \neq 0$, where $a, b \in \mathbb{R}$, and the binary operation $*$ on $G$ is given by $$(a, b) * (c, d) = (ac, bc + d)$$ Then, the identity element $e = ...
user avatar
6 votes
0 answers
64 views

When does a modified distributive law, like $a\cdot(b+c)=(a\cdot a)+(b\cdot c)$, allow polynomials to be written in the standard form?

Suppose we have two unary operations $f,g$, related by the law $$\forall a,\quad g(f(a))=f(f(g(g(a)))),$$ that is, $gf=ffgg$. We might then wonder whether any expression involving these operations can ...
mr_e_man's user avatar
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0 votes
0 answers
69 views

Is 2 some sort of identity (like 0 and 1 are)?

I am not a serious mathematician, and I am barely familiar with some of the topics of higher maths. From YouTube videos, I learned about the Graham's Number and Knuth's arrow notation, which is a ...
Parzh from Ukraine's user avatar
1 vote
0 answers
30 views

Proving associativity of binary operation

If $S = \{x\in\mathbb{R} \mid x>0\}$ with the binary operation $\star$ given by $$ x\star y = \sqrt{xy} $$ Then to show S is a group we of course need to show $(S,\star)$ satisfies the group axioms,...
altayir1's user avatar
0 votes
0 answers
45 views

Is there a well-defined set encompassing all possible operators in math?

The most elementary math operation always has two operands and an operator. For instance, the addition of two operands $a$ and $b$ can be represented as: $$a + b, \quad a, b \in \mathbb{C}$$ This ...
Aniket Shinde's user avatar
0 votes
0 answers
106 views

Left/Right Identity & Inverse Elements For Binary Operations

I'm having a difficult time in my Abstract Algebra course with the concept of left/right identity and inverse elements for a binary operation. On a recent assignment (which I will be altering slightly ...
upas's user avatar
  • 19
7 votes
1 answer
407 views

Possible group operations on a finite set

Suppose $X=\{x_1, x_2, \ldots, x_n\}$ is a finite set of $n$ elements. I learned that there are $n^{n^2}$ binary operations $*:X\times X \to X$ and $n^{n(n+1)/2}$ of them are commutative. I was ...
Nothing special's user avatar
2 votes
2 answers
173 views

Showing permutation does not change output of commutative operation (recursion theorem)

(In what follows, NB that $\mathbb{N}^\times$ is the positive natural numbers, and $\mathbb{N}$ includes 0.) I am working on Exercise 1 of Amann and Escher Analysis I, and the problem is essentially ...
EE18's user avatar
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1 vote
0 answers
48 views

Equivalent equational identities which are not alphabetical variants of each other.

Let our signature be that of a single binary operation $+$. Suppose $E$ and $E'$ are equivalent equational identities. Suppose also that neither $E$ nor $E'$ are equivalent to the trivial identity $x=...
user107952's user avatar
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1 vote
0 answers
38 views

Equational identities equivalent to the associative identity

This is a natural follow-up to my previous question, here: Is only the commutative identity equivalent to the commutative identity?. As usual, let our signature be that of a single binary operation $+$...
user107952's user avatar
  • 21.4k
0 votes
1 answer
61 views

How do I use lft cancellation to solve this problem?

Find a binary operation $m$ on $S = \mathbb Z$ which has an identity, is associative and satisfies: $$∀x, y, z ∈ S, m (x, y) = m (x, z) ⇒ y = z$$ but which does not form part of a group structure on $...
mlungisi blessing's user avatar
1 vote
1 answer
42 views

Identities equivalent to the reflexive identity

This is a follow-up to my previous question, here: A characterization of the identities which are equivalent to the trivial identity. As in that question, let our signature be that of a single binary ...
user107952's user avatar
  • 21.4k
0 votes
2 answers
128 views

Show that $\sigma$ is a bijection if and only $u$ is a unit.

The full exercise asks If $M$ is a monoid and $u \in M$, let $\sigma: M \to M$ be defined by $\sigma(a) = ua$ for all $a \in M$. Show that $\sigma$ is a bijection if and only $u$ is a unit. The ...
iwjueph94rgytbhr's user avatar
4 votes
0 answers
64 views

Is it decidable if a finite set of identities imply the commutative identity?

This is a follow-up to my previous question, here: Is it decidable if a finite set of equations have only trivial models?. Let our signature be that of a single binary operation symbol $*$. Suppose I ...
user107952's user avatar
  • 21.4k
1 vote
1 answer
159 views

Normalised binary exponential form

I am doing some mathematics, and I am currently stuck on something. I do not understand this part at all, expressing the result in normalized binary exponential form, the book also mentions ...
Alix Blaine's user avatar
-1 votes
1 answer
82 views

How to concatenate binary numbers without using their decimal values? For example 3 and 4 -> 34

This question describes exactly what I need to do! I am also doing this using electronic circuitry. The problem is, that the answer (which has been marked as a 'solution') only works for those two ...
TheMatrixAgent22's user avatar
0 votes
2 answers
88 views

Possible outcomes of sums

Let $\mathrm{S}$ be a set of distinct 20 integers. A set $\mathrm{T}_{\mathrm{A}}$ is defined as $\mathrm{T}_{\mathrm{A}}:=\left\{\mathrm{s}_1+\mathrm{s}_2+\mathrm{s}_3 \mid \mathrm{s}_1, \mathrm{~s}...
Snowball's user avatar
  • 1,023
1 vote
1 answer
69 views

Uniqueness of Binary Operations

I was doing an AMC question that used a binary operation, ?, that was defined as such: a?(b?c) = (a?b)*c , for all real, non-zero a,b,c, where * is normal multiplication. In addition, (a?a)=1 for all ...
Isaac Wachsman's user avatar
1 vote
1 answer
73 views

Can a vector with unordered components exist?

It seems like in order for a vector addition to be commutative, it needs to be defined in a "regular" manner, i.e. by adding matching vector components (because then the commutativity of ...
KKZiomek's user avatar
  • 3,875
0 votes
1 answer
67 views

Is this definition of commutativity correct?

Everywhere I see commutativity defined somewhat like this: A binary operator $*$ is commutative in $S$, if for any $x, y \in S$, the following property holds: $x*y=y*x$. That definition is, of ...
KKZiomek's user avatar
  • 3,875
5 votes
2 answers
124 views

Expected value of $3 \circ 3 \circ 3 \circ \dotsc \circ 3$, with $\circ \in \{+,-,\times,\div\}$

You are given an expression with $n$ 3's and $n-1$ $\circ$'s and you wish to evaluate the expected value of $3 \circ 3 \circ 3 \circ \dotsc \circ 3$, with $\circ \in \{+,-,\times,\div\}$. What I did ...
Joseph Bendy's user avatar

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