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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which X solves: $X \oplus 24 = X$ [closed]

which X solves the following equation? $X \oplus 24 = X$ I tried many numbers and nothing seems to solve it... Update: $X$ is a natural number and $\oplus$ is bitwise XOR
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What kind of operation is cube root extraction?

I came across this question in a random test and the correct answer was marked as "Binary Operation". I am pretty sure that to find the cube root of a number you only need that number alone ...
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2 answers
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Set $G=\{C\subset\mathbb{R}^n\}$. Is there a binary operation $\cdot$ such that $(G,\cdot)$ is a group?

Set $G=\{C\subset\mathbb{R}^n\}$. Is there a binary operation $\cdot$ such that $(G,\cdot)$ is a group? The biggest trouble I'm having is concerning the existence of inverses. A standard notion of &...
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Prove or disprove: there exists no binary operation $*$ on $\mathbb Q^+$ s.t. $(\mathbb Q^+,*)$ is not isomorphic to $(\mathbb Q^+,×)$

I am a first year college student who just started studying abstract algebra. I have been discussing the following problem with my friends at another university for a couple of days: Prove or disprove:...
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3 votes
1 answer
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Algebraic structure satisfying interchange law

Is there a name for an algebraic structure that satisfies a rule equivalent to the interchange law from category theory? Say, a set $A$ with associative (not necessarily commutative) operations $\cdot$...
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Is it necessary that a binary operation is commutative so as to have a two sided identity element? [closed]

Is it important that a binary operation is committed for having a two sided identity element in it
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Binary Relation definition in Set Theory?

Do binary relations have multiple definitions? For example equality: $$= \subset U \times U$$ $$(a,b) \in =$$ appears used typically as: $$a = b, =(a,b)$$ which would require a functional definition ...
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How many associative binary operations on the integers does $+$ distribute over?

I am interested in binary operations $\mid: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ which satisfy: Associativity: $a \mid (b \mid c) = (a \mid b) \mid c$ $+$ distributes over $\mid$: $(a \mid b) ...
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The given function $\phi$ is not an isomorphism because it is not surjective. But it's not injective either, right?

Here's the problem I did for homework from A First Course in Abstract Algebra, 7th Edition by John B. Fraleigh. I just want to check if my reasoning is correct on problem number 15 from Section 3: ...
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3 votes
1 answer
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XOR operation inner-product proof

Consider the binary power set $\underline{K}$ which contains binary strings of length $N$. Example: Let $N=3$, we then have $$\underline{K} = \big\{\{000\},\{100\},\{010\},\{001\},\{110\},\{101\},\{...
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3 answers
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Is addition really a binary function??

My class 12 maths textbook says addition is a binary function. But I had this experiment which caused a doubt. Addition is explained as a binary function because if you add more than two numbers you ...
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1 answer
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Field theory and distributive law.

How do any subset of a Field inherit the properties-commutativity,associativity and specially Distributive law over the same operationsas that of field?Is there any intuitive way to understand that ...
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How do we prove the following?

The expression is: a1 | a2 |⋯| an ≤ a1 + a2 +⋯+ an , where '|' denotes the Bitwise OR operator
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1 answer
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Binary inner-products with XOR operation

Consider the binary power set $\{\underline{K}\}$ which contains binary strings of length $N$. Example for $N=3$ we have $$\big\{\underline{K}\big\} = \big\{\{000\},\{100\},\{010\},\{001\},\{110\},\{...
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Is it necessary to check every possible triple of values to confirm a binary operation is associative?

Is it possible to use less than n^3 (where n is the cardinality of the set the operation works on)? That is, is there some way ...
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2 votes
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Reversing a complex boolean function

I am trying to reverse-engineer the DRAM address function of a memory controller. This is a function that maps a physical address to a DRAM bank. More formally, I am trying to find functions $f_i$ for ...
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Is there a named operation for the sum of a vector's elements?

If I have a vector $<a,b>$ is there a name or specific notation for adding the elements to create a scalar $c$, such that $c=a+b$? For reference, I want to use this to rationalize a notion I ...
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How many adders needed in order to implement $y=x^2$?

We are given a natural number $x\in\mathbb{N}^+$, and $x-1$ adders. An adder in our case would be a component with two inputs $x_1,x_2$ and a single output $y$ such that $y=x_1+x_2$. Our goal is to ...
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1 answer
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Does there existes a 'Tuple theory'?

When you search for set-theory on google, you can find many relevant results about it. Set theory defines many operations on sets like compliment, union, intersection etc. But searching tuple-theory ...
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-4 votes
1 answer
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Difference between binary division and its decimal division [closed]

Suppose I have one decimal number $23$ which decimal representations is $10111.$ Now $10111$ treated as dividend and divisor is $3$ which binary representations is $11.$ When $10111$ is divided by $11$...
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Is there any operation/tool/procedure that generalizes all the other operations?

As the title of the question suggest. Is there any tool that generalizes all mathematical operations, like adding, subtracting, dividing or even integrating, deriving or making a matrix transformation....
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2 votes
1 answer
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If $\pi: G \to H/K$ such that $\pi (g) = \varphi(g)K$, prove that it is well-defined.

Suppose that $(G, \cdot)$ and $(H, *)$ are groups. If $K \trianglelefteq H$, $G \cong H$ and the mapping $\pi: G \to H/K$ is defined as $\pi (g) = \varphi(g)K$, prove that it is well-defined. Here $\...
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-1 votes
1 answer
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Define a binary operation on {e,(12)}×{e,(123),(132)} so that it becomes isomorphic to $S_3$ [closed]

Since {e,(12)} isomorphic to $Z_2$ and {e,(123),(132)} isomorphic to $Z_3$ and gcd(2,3)=1, so, {e,(12)}×{e,(123),(132)} isomorphic to $Z_6$ which is not isomorphic to $S_3$. But can a binary ...
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Puzzle: binary number shrinkage on operation

Given a random binary number with n digits. Define operation P: count number of ones, suppose there are k ones in this binary number, then flip k-th digit(1 to 0, 0 to 1) counting from left(or right, ...
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Example of a set not closed under multiplication

It might be a stupid question, but can you give me some example of multiplication not being closed in some set? I could find a case in "addition"(e.g., a set of odd numbers is not closed ...
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3 votes
1 answer
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Does this property of two binary operations have a name?

I am wondering whether the following property of two binary operations $\diamond$ and $\star$ has a name. I haven't seen it listed in overviews of properties of binary operations, and I wouldn't know ...
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How do we add multiple binary bits with carry using boolean operators XOR and AND.

My question is similar to How do I add multiple binary numbers without using a partial sum?. For example, if we add two bits, a and b, then sum bit = a XOR b and carry bit = a AND b. Is there a way to ...
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0 votes
1 answer
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Boolean Logic - Conversion from AND to OR and OR to AND

is there a simple boolean equation that : Given Equals A+B AB AB A+B !A+!B !A!B !A!B !A+!B I figured that it would require to flip the sign, but then it lead me to this Given Flip Equals AB A !...
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Class of Operations With Verifiable Parameters

What is the proper terminology for operations that allow checking if a specific parameter/operand value was used? For instance, an imperfect example would be the concatenation operation: concatenate(&...
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1 answer
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Binary multiplication of 11001000 and 10011010 - Different answer?

I've run into a bit of an issue during my multiplication of these two binary values: $11001000\ (0.11001000 \times 2^2)$ and $10011010\ (0.10011010 \times 2^0)$. The answer I get when I did the ...
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In Polish/Prefix notation of binary operations, are all expression unambiguous w.r.t. which operator is applied first?

Take binary operator ~ and a chain like a ~ b ~ c. It is ambiguous as it can mean either of: ...
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-1 votes
1 answer
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What is the identity number for exponentiation/logarithm?

If the identity number of the addition/subtraction pair of operations is 0. and the identity number of the multiplication/division pair of operations is 1. what is the identity number of the ...
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Dividing with a bigger divisor

I have a task with binary operations 10 ÷ 10011 I have trouble in solving this equation. I tried searching but i can't find one with same given values (where divisor is bigger than dividend). If ...
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4 votes
1 answer
56 views

Untyped λ-calculus: proof that for any binary relation $R \vDash \lozenge \Rightarrow R^* \vDash \lozenge$

I'm currently in the process of reading Barendregt's "The Lambda Calculus - Its Syntax and Semantics" (1985 revised edition) and I've stumbled across a lemma whose proof I can't quite ...
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-2 votes
3 answers
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Distinguishing the $+$ in $\mathbb R^n$ from the $+$ in $\mathbb R$ (where NOT distinguishing MAY INDEED cause confusion)

Edit based on 2 answers given: The context is that this question was originally part of another question. I believe the lack of distinguishing between the 2 $+$'s is because the instructor wants to ...
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Is integration unary or binary operator?

Is integration unary operation or binary operation? If we think about numbers, then integrals are interated binary operation. But if we think in terms of functions, then integrals seem unary operators....
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3 answers
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Are there commonly used symbols for the negation of implications?

Exhaustively, there are 16 possible truth tables for two propositions. If we interpret the values in the rows as binary digits, we can conveniently use the resulting hexadecimal digit as the ...
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0 answers
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Booth algorithm for multiplying signed numbers

The algorithm of the Booth for multiplying signed numbersrs of fixed complement representation decimal point by 2 is implemented by multi-consecutive digit stain control of the multiplier, so after ...
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1 answer
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All combinations for $x_i+y_j$ in a finite sets of real numbers

Consider $m, n \in \mathbb{N}$ and $\{x_1, x_2, \cdots, x_n\},\{y_1, y_2, \cdots, y_m\} \subset \mathbb{R}\setminus \{0\}$. Question. How many possible combinations we can form with elements of the ...
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1 vote
1 answer
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Multiplicative $\mathbb{Z}^{*}_p$

Question: How to prove that $\mathbb{Z}^{*}_p$ is a group under multiplication operation where $p$ is prime. Proof: We define $$\mathbb{Z}^{*}_p=\{[a] \in \mathbb{Z}_p: [a] \neq [0]\}$$ 1. For well-...
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2 votes
0 answers
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Fewest applications of associativity

By repeatedly applying the basic associativity law $(x+y)+z = x+(y+z)$, one can get from any one expression with binary addition to any other with the variables in the same order. Specifically, given ...
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2 votes
1 answer
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Help with binary division problem dividing $1001011010000$ by $1001$

I have this binary division practice problem, I am not sure how to deal with a leading 1 when pulling down the next bit. Here is my process, I'll try to make it clear to where my confusion comes about,...
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2 votes
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Is there a standard name for this property of ordered pairs of binary operations?

I know that the distributive property of ordered pairs of binary operations is well-known. However, I have thought of a new property of ordered pairs of binary operations. Let $+$ and $*$ be the ...
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5 votes
1 answer
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Are there precedence (BODMAS kind of) rules in set operations?

Let's say we have three set's $A, B, C$ with elements $\in \Bbb Z$ . If there is an operation $A \setminus (A\cap (B \setminus C)^\mathsf{c})\cup(B\cap C)$ let $x=(A\cap (B\setminus C)^\mathsf{c})$ ...
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3 votes
1 answer
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How to show associativity with set of only two elements.

Considering the set $S= \{-1, 1\}$, I need to show if associativity holds under binary operation multiplication. We used to take any three elements to verify the associative property, but here only ...
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0 votes
1 answer
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Is closure enough for an operation to be chained?

Say we have a set S and a binary operation $\star$ under which S is closed. Is this enough for us to derive an arbitrary (possibly infinitary) operation $\star$ in which the order of operations ...
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Non-contradictory axiom system for a binary operation

Suppose we want to define a binary operation $\otimes:\mathbb{N} \times\mathbb{N} \rightarrow \mathbb{N}$ on a ring $(\mathbb{N},+,\cdot)$ with an arbitrary system of axioms. The axioms may be given ...
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XORing two fractions of integers

I was wondering if it is possible to apply the XOR operator between two fractions of integers. 2 ⨁ 13 = 0010 ⨁ 1101 = 1111 = 15. ...
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Number of binary operations isomorphism with 4 elements

Let $A = \{1, 2, 3, 4\}$. What is the number of binary operations $*$ defined on $A$, such that $(A,*)$ is a group isomorphism to $(\Bbb Z_4,+)$, and the order of $3$ is $2$ in this group ? I tried ...
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How to find cyclic subgroups of $S_4$ with certain numbers of elements [duplicate]

In the group $S_n$ consisting of the set of all bijections from $\{1,2,...,n\}$ to $\{1,2,...,n\}$ with a binary operation $◦$ denoting composition of functions. I am asked to find cyclic subgroups of ...
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