Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
37 views

Is there a set in which division of 0 by 0 is defined?

The reason I ask this is that I've discovered that, even though they don't satisfy all field axioms, there are sets called projectively extended real number line and Riemann sphere, which are ℝ∪{∞} ...
0
votes
0answers
19 views

Function/Operation Definitions

If I want to define a function/operation for equivalence classes, is it permissible to stipulate contingencies based upon element characteristics or does dependency on the choice of element ...
4
votes
0answers
53 views

An $m$-ary function that represents all $n$-ary functions

Motivation It is well-known that any binary operator $*$ on the boolean ring $\{0,1\}$ can be represented using only one of the $\operatorname{NAND}$ and $\operatorname{NOR}$ operators. For example,...
-2
votes
0answers
35 views

Proof of closure for x * y = (x^3 + y^3)^⅓

Let * be defined on ℝ as: $$ ∀ x, y ∈ ℝ: x * y = (x^3 + y^3)^⅓ $$ I have tried to prove $(x^3 + y^2)^⅓$ is a member of ℝ to show closure, but I'm not sure how. Do I need to prove it or is it just ...
1
vote
0answers
31 views

Help Me Understand the Generalisations of Commutativity and Associativity for Unions

Introduction I can prove the generalisation of commutativity and associativity of unions, but I do not understand how they are generalisations of commutativity and associativity. I think it would ...
0
votes
2answers
11 views

Inverse operation of xor

If x = a xor b, given the values of x and a can we find b? In other words, which function can be applied on both sides in the equation to get the value of b?
0
votes
0answers
17 views

Floating Point to Decimal Conversion

If a floating-point number is stored in one byte such that the first bit is the sign, the next three bit represent the exponent in excess-4 notation, and the last four bits represent the mantissa, ...
-1
votes
1answer
38 views

Is $f(g,h) = g^2$ a binary operation on a group $G$? [closed]

If $(G,\cdot)$ is a group, then $f : G \times G\to G$, defined by $f(g, h) = g^2$, is a binary operation on $G$. Is it true or false and why?
0
votes
1answer
24 views

Image of a binary relation

I know it might seem a stupid question but I need some clarification. From my notes: "a binary relation $R$ associated to each element $x$ of $X$ some elements $y$ of $X$. We denote by $R(x) = \{y \...
0
votes
1answer
44 views

If $x,y \in \mathbb{N}$ then $x+y=0 \iff x=y=0$

Let $x,y \in \mathbb{N}$. The operation $(+)$ is defined by: $$x+0=x$$ $$ x+(y+1)=(x+y)+1$$ Then prove that $x+y=0 \iff x=y=0$. The second implication $x=y=0 \implies x+y=0 $ is simple and ...
4
votes
2answers
107 views

Commutative Semigroup

Let $S$ be a Semigroup with the two following properties, $(1):$ for all $x$ in $S$ we have $x^3=x$ $(2):$ for any $x,y$ in $S$ we have $xy^2x=yx^2y$. Then prove that this Semigroup $S$ is ...
1
vote
1answer
40 views

Check the properties of the following operation defined on R

An operation is defined on $\mathbb{R}$ such that for every $x,y \in \mathbb{R}$, $x \ast y=\sqrt{x^2+y^2}$. I was checked some of the basic properties like commutativity, associativity and whether ...
0
votes
1answer
46 views

Is it possible to reduce an algebraic function to $1$s and $0$s?

I want to know where in this process I'm going wrong. Perhaps it's not even a valid thing to do...? Take a well-behaved function such as $f(x)=x \sin 2x$. I want to turn this into a new function $g$ ...
1
vote
3answers
35 views

Set $S= \mathbb{Q} \times \mathbb{Q}^{*}$ with the binary operation $(i,j)\star (v,w)=(iw+v, jw)$

Consider the set $S= \mathbb{Q} \times \mathbb{Q}^{*}$ with the binary operation $(i,j)\star (v,w)=(iw+v, jw)$. (a) Show that the binary operation is closed. I said the operation is closed under ...
1
vote
1answer
34 views

isomorphism in a product

$x*y=\frac{x+y}{1+xy} , x,y\in(-1,1).$ Calculate the value of $ \frac{1}{2}*\frac{1}{3}* \cdots *\frac{1}{1000}.$ I tried a lot of functions but I don't know how to find a good isomorphism and do the ...
1
vote
1answer
43 views

Ambiguity with parentheses multiplications [closed]

I was recently shown the equation $6 \div 2(1 + 2) = ?$, and it was disputed whether this equation equals $1$ or $9$. To solve for $1$: $$ 6 \div 2(1 + 2) \\ 6 \div 2(3) \\ 6 \div 6 \\ 1 $$ To ...
0
votes
0answers
20 views

Proper class of operations

Usually, a collection of operations forms a set. But I've heard of an example where it is (must be) a proper class. Namely, that $\mathbb{CompHauss}$ becomes a variety if proper class of operations ...
4
votes
1answer
88 views

Name for generalization of property: $f^n(x) \ne x$ for all $n > 0$

I am curious about how to specify with standard terminology that a certain function is non-repeating, in the following sense: In the simple case of a unary operation $f: X \to X$, this property would ...
0
votes
1answer
21 views

Which integers have inverses with respect to lcm operator, a*b=lcm (a,b) [closed]

I know that the identity of this operator is 1, but am not sure about the inverse.
1
vote
1answer
57 views

If symmetric difference leaves a set unaffected, the second set is empty.

To prove: If $A \triangle B=A$, then $ B= \emptyset$. This seems simple enough, as an idea. I mean if the set $B$ is anything but empty, $A\triangle B$ would contain more or less than simply $A$, ...
14
votes
4answers
1k views

Do binary operations need to be surjective functions?

Let $\star$ be a binary operation on the set $S=[0,1]$ defined to be $$\star : [0,1] \times [0,1] \to [0,1] $$ $$\text{where } a \star b = \text{min}\left(\frac12 a , \frac12 b\right) $$ From ...
1
vote
0answers
16 views

Binary operation of addition on reals is uniformly continuous, whereas multiplication is not (proof-verification)

I would like to ask whether the following proof is correct and, in particular, why the step (*) of this proof is valid? The problem is to show that the function $+: \mathbb{R}^2 \to \mathbb{R}, (x,y)\...
0
votes
0answers
32 views

How to find the number of msb bits common between two binary numbers

Im trying to find the number of bits common between two binary numbers starting from MSB -> LSB. For example, I'm taking a set of binary numbers of 4 bits each i.e., 0 -> 15. I'm trying to find the ...
0
votes
0answers
32 views

How define formally a user-defined binary operation?

Let $x \in \{0,1\}^{|\mathcal{S}|}$ and $k \in \{0,1\}^{|\mathcal{R}|}$, where their elements are indexed by the index sets $\mathcal{S}$ and $\mathcal{R}$ respectively. The index sets satisfy $\...
7
votes
0answers
165 views

Associative, non-commutative, non-trivial, analytic binary operation

There was a question whether associative, but non-commutative binary operation over the real numbers exist. A trivial answer is the binary operation $x\circ y = x$ or $x\circ y = y$. As a followup, ...
8
votes
2answers
201 views

Associative, non-commutative, nontrivial operation on the real numbers

This MSE question asks about binary operations on the real numbers which are associative, but not commutative. Two answers are given: The operation $\circ$ defined by $x \circ y=x$. Letting $f:\...
1
vote
0answers
36 views

Generalised Binary Operation?

In algebra, we study sets with binary operations satisfying certain properties. However, sometimes that the calculation is undefined for some ordered pairs, such as $a\div 0$ for any complex number $a$...
0
votes
1answer
72 views

Finding a Recurrence Relation for a binary string with n digits that do not contain 000

Consider binary strings with $n$ digits (for example, if $n=4$, some of the possible strings are 0011, 1010, 1101, etc.) Let $z_{n}$ be the number of binary strings of length n that do not contain ...
1
vote
2answers
80 views

Is there a fundamental mathematical function that requires 3 inputs or more?

So a mathematical operation can be represented as a function that maps inputs to outputs. For example "sin(x)" is a function that maps 1 input to 1 output, and "a + b" maps 2 inputs to 1 output. My ...
0
votes
2answers
41 views

Example of a semigroup satisfy some properties

I want to find an example a finite semigroup $S$ and $K \subseteq S$ satisfy the properties For any $a,b \in K$, we have $a,b \in \langle c \rangle$ for some $c \in S$. $K$ does not hold closure ...
1
vote
2answers
79 views

How many associative binary operations are there on a 2 element set?

We can easily find commutative binary operations on a 2 element set from the truth table (if ab=ba then the operation is commutative, thus there are 8 commutative binary operations in a 2 element set)....
1
vote
2answers
74 views

Define $*$ on $\mathbb{Z}$ by $a*b = a+b$. Show $*$ is a binary operation on $\mathbb{Z}^+$.

Let $S = \mathbb{Z}^+$. Define $*$ on $\mathbb{Z}$ by $a*b = a+b$. Show $*$ is a binary operation on $\mathbb{Z}^+$. In our course on Abstract Algebra our book says the following in it. $*$ is a ...
0
votes
1answer
48 views

how to combine 2 sequences of binary numbers

l have two binary sequences, K1 and K2 K1 : 1 0 1 K2 : 0 0 1 l would want to perform binary operations on both sequences K1 and K2 so that one binary number is the output for example the output is ...
-1
votes
1answer
81 views

Is inverse possible for getting identity element ($e$) for binary exponentiation operation? [closed]

I am assuming two domains below: integers ($\mathbb{Z}$), & rationals($\mathbb{Q}$); with common identity element $1$. The binary exponentiation operator yields for the simplest case of two ...
0
votes
3answers
70 views

Unary Operations Vs Unary function

Im very confused with unary operations. I have read from WikiPedia and many other sites what unary operations are. Specifically... In mathematics, a unary operation is an operation with only one ...
0
votes
1answer
39 views

Need the name of algebraic structure suitable for this

If I want to prove that addition and multiplication are internal operations in the rational domain Q. What algebrac structure I need to prove that (Q,+, *) will be? ...
0
votes
3answers
54 views

Does general commutativity require associativity?

There is a theorem that if an operation is associative on 3 elements from a set, it is associative on any number of them. This property is called general associativity. Similarly, there's a theorem ...
6
votes
2answers
102 views

Check if $G$ is group or not

Let $G=\{0,1,2\}$ define $*$ on $G$ such that $a*b=|a-b|$. Check if $G$ is a group Edit : As @egreg correctly pointed out if you draw the cayley table then one can directly see it is not a group , I ...
2
votes
0answers
48 views

Is there a measure for how associative a binary operator is?

For some binary operation defined on a set, it is of interest whether the operation is associative or not (among, of course many other things). Is there a measure (and accompanying theory) that ...
3
votes
4answers
230 views

Well definedness of factor group multiplication

I have been reading on this site and the internet, and I do not quite understand the comments which are being made with respect to the problem of proving that factor group multiplication is well ...
3
votes
1answer
58 views

Equivalent algebraic theory with at most binary operations

Given an arbitrary (one-sorted) algebraic/equational theory $\mathbb{T}$, is it possible to devise an equivalent (one-sorted) algebraic theory $\mathbb{S}$ such that the signature of $\mathbb{S}$ ...
1
vote
0answers
47 views

Prove associativity for the following space.

Let $X$ be the set of all $A\subset\mathbb Z$ that are bounded above; that is, $A\in X$ iff $A\subset\mathbb Z$ and $\exists\max A$. Define the following operation, as the sum of two such sets: $A\...
0
votes
0answers
44 views

The rational sum

Let $x=\frac{a}{b}$ and $y=\frac{c}{d}$ be two rational numbers with fractions in simplest form. Can we express $\frac{a+c}{b+d}$ in terms of $x$ and $y$ only. Also, Can $\frac{a+c}{b+d}$ be a binary ...
1
vote
1answer
27 views

Proving a binary operation is not associative given a latin square

Given a Latin square how would one tell if the operation is associative without trying every combination? Or is there something to look for that would at least limit the amount of combinations I have ...
-1
votes
1answer
31 views

How to Create a Relationship between 3 binary vectors

I have two process that l will perform. In the first process l have the following binary vectors that l would want to create relationship between them : A, B, C and the resulting F Formulating a ...
2
votes
1answer
160 views

Binary Operations (Commutative and Associative).

Let $*$ be a binary operation on $\mathbb R$ given by $x*y = (x^{1/3}+ y^{1/3})^3$. Determine if it is commutative and associative. I know how to prove it is commutative. \begin{align*}x * y &...
0
votes
1answer
50 views

What are the binary operations of a ring?

I'm just reading through the definition of a ring and it says that "A ring is the triple (R,+,•) where R is a set and +,• are binary operations". Does • imply multiplication since for R to be a ring ...
1
vote
1answer
59 views

Formal Model of Immutability

Wondering if there is a formal model / algebra / etc. of immutability. This comes up in functional programming with persistent data structures, but I haven't seen any pure math related to it. For ...
0
votes
1answer
59 views

Number Theory Help!

Numbers in $\mathbb{G}$ are ordered pairs of integers, i.e. (a, b) ∈ $\mathbb{G}$ if a ∈ $\mathbb{Z}$ and b ∈ $\mathbb{Z}$, and binary operations ⊕ and ⊗ are defined in $\mathbb{G}$ by $$(a, b) ⊕ (...
3
votes
1answer
355 views

How to recover $k$ lost items in binary data $x_1,x_2,x_3 \dots,x_n$?

$$x_1,x_2,x_3,\dots,x_n$$ where they all are same size binary data. If we lose one of them in series and if we want it to recover, We just need to store $y_1$ that is same size like the items in ...