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.
915 questions
0 votes
1 answer
36 views
Is every serial relation a generalized divisor relation?
First, some preliminary definitions. A serial relation is a binary relation $R$ on a set $S$ where this property holds: $(\forall x)(\exists y)xRy$, where the quantifiers range over $S$. Now, let $*$ ...
- 24.9k
2 votes
2 answers
96 views
Does this define a cross product on the bivectors of 6-dimensional space that satisfies the Jacobi identity?
Background: The vector cross product (in 3-dimensional space, $V = \mathbb{R}^3$) satisfies the Jacobi identity, and so it is a bracket operator that makes $V$ a Lie algebra. We can also look at the ...
- 501
0 votes
0 answers
48 views
Grey Code Index from a Point
I've been interested in Binary Logic and various functions of Binary numbers, and I've come across Z-order curves and Gray Codes. A Z-order curve is the curve that results from the splitting of a ...
-2 votes
2 answers
125 views
Is there a function that satisfies both logarithmic and exponential addition identities?
My objective for this post is to set parallels between logical, set, and numerical unary and binary operations. That is, I want to relate: Union($\cup$) --> Or($\vee$) --> Addition (+) ...
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1 vote
1 answer
154 views
Is there a name for the operator $⊞$ defined as $a ⊞ b = \sqrt{a^2 + b^2}$?
It's not uncommon to have a meaningful relationship c² = a² + b². This happens in both geometry (Pythagorean theorem) and statistics (sources of variance) at least, and I think I've seen it elsewhere. ...
- 193
0 votes
0 answers
39 views
Why is the MSB equal to 1?
I'm figuring out the value of $1110_2 - 11_2$ and here are my steps: $11_2=0011_2$ $1100_2$ - 1's complement $1101_2$ - 2's complement $1100_2 + 1101_2 = 11011$ Answer: $1011_2$ after dropping ...
- 1,028
8 votes
2 answers
384 views
How do I cook my food efficiently
What is the optimal strategy for flipping N pork tenderloins? The Setup Today, I decided to pan-sear some pork tenderloin. I sliced the piece into $N=15$ roughly identical cylinders and arranged them ...
- 252
-1 votes
1 answer
210 views
Proving that folding a list with any commutative and associative operation yields a result independent of fold order
Setup $\DeclareMathOperator{\lcm}{lcm}$ Suppose that $*$ is a binary operation between $a$ and $b$ that is both commutative and associative. Some examples include: Addition ($a+b$) Multiplication ($a ...
- 119
1 vote
2 answers
95 views
What are the semantical pre-requisites for an operation being associative?
I beginning to start math from the start again and I'm trying to understand things more intuitively and I'd like some help to understand associativity. I can understand it as the characteristic of an ...
- 11
-1 votes
1 answer
80 views
Convolution of Random Variables [closed]
The distribution of a sum of two independent random variables is equal to the convolution of their probability distributions. However, random variables and their probability distributions are often ...
- 2,184
2 votes
1 answer
75 views
Openness of a set under a binary relation?
Considering $(X,T)$ to be a connected topological space, $f$ being a continuous function from $X \to X$ and the binary relation being irreflexive, transitive, complete and open. $X^2$ has the product ...
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2 votes
1 answer
44 views
Do antitonic, subidempotent functions have to be constant under a total order on $X$ ($|X| > 1$)?
Assuming $\leq$ is a total order, in other words a reflexive, transitive, anti-symmetric, connected (every pair is comparable) binary relation, on a set $X$ of cardinality $\geq2,$ does the condition $...
- 1,661
3 votes
1 answer
127 views
Commutative and associative binary operations over $\mathbb R$
Are there examples of commutative and associative binary operations over $\mathbb R$ that cannot be expressed as $x \oplus y = f^{-1}(f(x)+f(y))$ where $f$ is an invertible function $f:\mathbb R\...
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0 votes
0 answers
26 views
System of Linear Equations over Binary Field and described by Circulant Matrix
Consider a system of linear equations over the binary field (either homogeneous or inhomogeneous), which we can denote in matrix form as $Ax = b$ for some $b\in \mathbb{F}_2^n$ and $A\in\mathbb{F}_2^{...
- 399
0 votes
1 answer
49 views
How to prove that each commutative hyperation operation is distributive over the previous?
There exists a series of commutative hyperoperations with the defining trait that each operation is distributive over its predecessor. The operations are defined recursively, taking $$a \times_{n} b = ...
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