Mathematics Stack Exchange
Mathematics 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.

Note: Whether or not a given operation is a binary operation can depend on the set. For example, subtraction is not a binary operation on $\mathbb{N}$ but it is on $\mathbb{Z}$.

There are many objects in abstract algebra which require binary operations as part of their definition. These include: magma, semigroup, monoid, quasigroup, ring, and field.

An $n$-ary operation on $X$ which is a map $\ast : X^n \to X$. A binary operation is the special case $n = 2$.

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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: $(\mathbb{Z}_{n}$ , $\times_{n})\space$ s.t. $\space a*b\space =$ …
Benjamin R
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How to express the min operator as a binary operator

I'd like to use the $\min$ operator as a binary operator that returns the lowest of two given numbers. I'm not sure if this is the correct use of it, or if I should use something else. $\text{Given four integers, }A,B,C, \text{ and }D \text{, take…
fdisk
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How to create a new binary operation on a same set?

I was studying binary operation on a set. Then the following question came to mind. I tried to find an answer. also searched in website but could not get any satisfactory answer. the question is: is it possible to define two distinct binary…
KON3
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Computing Number of Possibilities

I'm asking this question in the context of a program I'm trying to write, but the problem I'm having seemed to be more of a mathematical one. (Also, I'm not quite sure what tags should be applied to this question, so please feel free to edit…
kingsfoil
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Binary operation on empty set?

Can we techniclly declare a binary operation on an empty set? Since binary operation does an action on some objects (which empty set dot have)... Thanks.
user144228
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Structured chain of functions

It is a trivial assertion that the binary operation $g(x,y)\mapsto xy$ is associative and also distributive over the binary operation $f(x,y)\mapsto x+y$. We say that $f\leadsto g$ if this occurs, where all the binary operations in consideration are…
user118228
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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 that if an operation between pair that "behaves"…
Andrea Miele
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error in textbook exercise regarding binary operations?

The following exercise of from Guide to Abstract Algebra by Carol Whitehead, 1st Edition 1988. Let $\bullet $ denote a binary operation on a non-empty set $S$. Suppose that $\bullet $ admits a left identity $e$ and a right identity $f$. Prove that…
Penelope
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Can this binary-expression be simplified?

Let $p = (a \land b) \lor (c \land d)$ where $\land$ and $\lor$ are bitwise AND and OR operators respectively. Can $p$ be simplified?
Anant
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Binary operation on the set of first n positive integers

Let $S$ be the set of the first $n$ positive integers. Suppose we have a binary operation @ that takes $a, b \in S$ to some $a @ b \in S$. Given that: 1 @ x = x @ 1 = x x @ (y @ z) = (x @ y) @ (x @ z) Prove or disprove: @ is associative I can't…
Hrittik M.
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Prove the distributive law for maximum and minimum operation

Given $A=\{1,2,3,4\}\subseteq \mathbb{N}$ and define the operation on $A$ as below \begin{eqnarray} a\oplus b&=& \max(a,b)\\ a\otimes b&=& \min(a,b) \end{eqnarray} for all $a,b\in A$. Prove $(a\oplus b)\otimes c=(a\otimes b)\oplus(a\otimes c)$ for…
Ongky Denny Wijaya
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Question about order of operations convention and alternative

I'm (trying) to make my own code parser that evaluates expressions. I've done some reading about this topic and seen the various mnemonics that school children are taught, such…
Zebrafish
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Ambiguity with parentheses multiplications

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 solve for $9$: $$ 6 \div 2(1 + 2) \\ 6 \div 2 \cdot 3…
Aly
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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 question is is there a function that requires a…
Parkman217
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Doubt regarding renaming of variables.

Problem: Consider a binary operation '*' on the set $S$. It is given that $(a*b)*a = b, \forall a,b \in S$. PT: $a*(b*a) = b \forall a,b \in S$ My attempt: Given : $(a*b) *a = b \forall a,b \in S$...........1) Or $((a*b)*a)*(a*b) = b* (a*b)$ Let $X…
Lelouch
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