# Tagged Questions

A binary operation on a set $X$ is a map $\ast : X \times X \to X$. Usually, we denote $\ast(x, y)$ by $x\ast y$.

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### Proving an idempotent binary operation where $(x\ast y)\ast z=(y\ast z)\ast x$ is commutative

Let $S$ be a set and $\ast$ be a binary operation on $S$ satisfying 1) $x\ast x=x$ for all $x\in S$, 2) $(x\ast y)\ast z=(y\ast z)\ast x$ for all $x,y,z \in S$. Show that $x\ast y=y\ast x$. I ...
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### Why is Nim solvable with the xor operator?

In the game of Nim, played with two players, if you have $n$ stacks of stones (where you can take any number of stones from a single pile each turn), losing positions are ones where the xor of the ...
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### Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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### Determine the operation based on the conditions given below

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
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### an operation using pit operators [closed]

Hello all i have some numbers and i make some operations in them which give me a number i save in a file so that's the codes ...
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### ⋇ “Division Times” operator in Unicode (U+22C7)? [duplicate]

I found this maths operator in Unicode: ⋇ It is called "Division Times" (U+22C7). Does it behave like ±? For example: 3 ± 2 means it is an ∈ {1, 5}. So 3 ⋇ 2 means it is an ∈ {1.5, 6}?
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### Can the state of a system after applying the operation “absolute value” be got back using elementary operations or transformations?

Take the operation or transformation "addition". You can get back the original state of the system by doing the opposite operation, i.e., "subtraction". But, if the operation is "absolute value", you ...
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### solving an XOR matrix

I'm working on a somewhat-unique linear algebra problem arising from XORing files together in order to encode them, and then figuring out how to subsequently recreate the original files from the ...
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### Binary Decision Trees

I know the basics to a binary decision tree, but this problem has me a little stumped, and I'm looking for some verification on my ideas. The problem is: "Create a binary decision tree that reflects ...
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### Proving that Z with the binary operation is a monoid?

Let $*$ denote the binary operation defined on the set $\Bbb Z$ of integers, where $$x * y = 3xy - 5x - 5y + 10$$ for all integers $x$ and $y$. Prove that $\Bbb Z$, with the binary ...
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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 ...
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### Characterization of elementary arithmetic operators to explain certain properties in programming languages

In the LISP-like family of programming languages, the four elementary arithmetic operators behave differently: + and * can take ...
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### Induction with associative binary operation

Let * be an associative binary operation on a set 'A' with identity element e. Let 'B' be a subset of 'A' that is closed under *. Let b1, b2, b3, ... bn ∈ B. Prove that b1 * b2 * b3... bn ∈ B. ...
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### Simple example of idempotent but not commutative nor associative binary operator?

Is there a simple example of a binary operation that is idempotent, but not commutative nor associative?