# 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 ...
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### 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 ...
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### 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, ...
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### 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 $A$ ...
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### 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 ...
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### 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 $+$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...