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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Solving for x in binary operations of a group

Given the equation, of a group A. Determine x in terms of j,k,l. $$k∗x∗x = j∗l$$ Given that we know that its a group, we know that the binary operation has an identity, each element has an inverse and it is associative. We can add $k^{-1}$ to the…
A. Harold
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Switching operands in a binary operation

I know it is a very simple question and maybe a waste of time, but I wanted to know if there is any mathematical name for the act of changing the order of the operands in a binary operation. That is, going from $a * b$ to $b * a$. Thanks.
Nada F.
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Operation with two identities?

I have read that an operation can have at most one identity and fully understand the proof, however what if I define an operation $*$ on $\mathbb Q$ as follows? $x*y=|x \times y|$, $\forall x,y \in \mathbb Q$ Surely both $1$ and $-1$ are…
RedG
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Give an example of a set with two binary operations, addition and multiplication, in which we have left distributivity but not right distributivity

Give an example of a set with two binary operations, addition and multiplication, in which the left distributive law holds but the right distributive law does not hold. I.e.: $$a(b+c)=ab+ac\text{, but }(b+c)a=ba+ca.$$ If this is not possible, then …
user333083
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Binary floating point subtraction

(In binary environment) 0.100011 * 2^6 - 0.111001 * 2^3 = 0.100011 * 2^6 - 0.000111001 * 2^6 = 0.100011000 * 2^6 + 1.111000111 * 2^6 (convert left part into 2's complement) = 10.011011111 * 2^6 However, answer was 0.011011111 * 2^6 Where did the…
구마왕
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I need my work checked(properties of operations)

Let $x * y = |x + y|.$ $x * y = |x + y| = |y + x| = y * x,$ so $*$ is commutative. $(x * y) * z = ||x + y| + z| = |x + |x + z|| = x * (y * z),$ so $*$ associative. $x * e = |x + e| = x,$ so $e = 0.$ Further, $e * x = |0 + x| = x.$ So, $*$ has an…
AbstraktAlgebra
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Show that $x$ is commutative and $+$ is associative for these binary operations on $\mathbb{R}^2$

\begin{align*} + : \mathbb{R}^2\times\mathbb{R}^2 &\rightarrow \mathbb{R}^2\\ ((a,b),(c,d))&\mapsto(ad+bc,bd)\\[1cm] \times: \mathbb{R}^2\times \mathbb{R}^2 &\rightarrow \mathbb{R}^2\\ ((a,b),(c,d))&\mapsto(ac,bd) \end{align*} Questions: a)…
Karl
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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 $e = (g, h)$, is such that $$(a, b) * (g, h) = (ag,…
user1236748
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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 notation allows for the selection of any complex…
Aniket Shinde
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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 under addition) but am struggling to find an example for…
Rowing0914
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Question on binary operations

A question defines A = {1,2,3,4,5,6} and a binary operation * such that $a*b=r$, where r is the least non-negative remainder when the product $ab$ is divided by $k$. Find k for * to be a binary operation. The answer is five, apparently, but; 1:…
harry
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Prove or disprove that if a and b have inverses with respect to ∗, then so does a∗b (where * is an associative binary operation with an identity e)?

This is part 2 of a question. Part 1 I was proving that the inverse of an element $a$ is unique with respect to $\cdot$ and I think I solved that one. Part 3 of the question is the opposite of part 2: Prove or disprove that if $a\cdot b$ has an…
Anna Kingsley
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How to find a pair of number (A,B) whose XOR is N such that B is smallest possible

What I'm trying to figure out here is, if there is any way that I can find such a pair of integer such that one of it is the smallest of any number occurring in any pair of number which satisfies A XOR B = N
Mohil Khare
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How to prove $R^k=R^{k+1}$

$R$ can be any binary relation on $S$. $R^0:=I=\{(x,x):x\in S\}$, and $R^{i+1}:=R^i\cup(R;R^i)$ for $i\geq 0$. [Here, $(R;R^i)$ denotes the composition of relations - ed.] If $|S|=k$, explain why $R^k=R^{k+1}$. I think it should prove if $(a,b)\in…
LIQI JIANG
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How to prove that an operation is binary?

I have been trying to learn binary operations and have not been able to understand how to prove that an operation is binary. For example: Show that $*:\mathbb R×\mathbb R→\mathbb R$ given by $(a,b)\mapsto a+4b^2$ is a binary operation. Now the…
JakePaul
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