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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Counterexample to show the map $a \mapsto -a$ is not a binary operation

The text in Dummit and Foote on pg-$16$ says: $-$ (usual subtraction) is a non-commutative binary operation on $\mathbb{Z}$, where $-(a, b) = a-b$. The map $a \mapsto -a$ is not a binary ...
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21 views

Properties of binary relations

I am so lost on this concept. We are doing some problems over properties of binary sets, so for example: reflexive, symmetric, transitive, irreflexive, antisymmetric. This particular problem says to ...
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17 views

Essential Prime Implicants and Minterm Expressions

I have an exam for a university course shortly, and upon reviewing one of my assignments I have come to realize that I don't understand why I have lost marks/how to do a couple of questions. Hopefully ...
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Binary remainder not equal to the decimal remainder

I am having a weird result. I am dividing the binary number $10101010100000$ by $10011$. In binary division. I get $R= 0100$ which is 4. However, If I consider the decimal representation of the ...
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32 views

Is the closure axiom necessary for algebraic structures defined via a binary operation?

Numerous algebraic structures are often defined as a set $X$ equipped with a binary operation $f:X\times{X}\rightarrow {X}$ that satisfies some set of axioms. Since the image of $f$ is always in $X$ ...
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31 views

Can all operations on multiple elements in mathematics be reduced to binary operations?

To take a very simple example: imagine the sum 1 + 2 + 3 + 4. You can do this one step at a time: 1 + 2 = 3, then 3 + 3 = 6, then 6 + 4 = 10. It does not matter how long it takes you to move from one ...
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41 views

Function, Relation, Operation and Cartesian Product

An operation is a kind of function. A function is a kind of relation. A relation is a subset of a Cartesian product. A Cartesian product is an operation. Back to 1. It seems to me that there's ...
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10 views

Evaluating the decimal equivalent of binary numbers in; sign and magnitude, one's complement and two's complement

For example, i have this binary number : 1011 1101 Now i wish to evaluate the decimal equilant using sign and magnitude, one's complement and two's complement. Now for sign and magnitude, i know the ...
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Explanation of the algorithmic form

Booth's multiplication algorithm is a multiplication algorithm that multiplies two signed binary numbers in two's complement notation.The core of the algorithm is the replacement of a string of $1's$ ...
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37 views

Probability of specific XOR value (hints only please)

I have a set with $5$ unique elements chosen randomly from the set of six-bit binary numbers, excluding $000000$. I take the power set of this set (i.e. the set of all subsets) without the empty set, ...
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12 views

Inequality bounds of binary variables

I came across a proof in AdaBoost paper (page 14/132). I couldn't follow a derivation at the left bottom of the page. It simply requires the below statements (i and ii) to be true: For $i=1,...,T$ ...
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Binary Operations, Associative Operations

I'm stuck on this question, please help. The binary operation $*$ is defined on $z$ by $x*y=xy-x-y+c$ for all $x, y, c$ belonging to $\Bbb Z$, $c$ is a constant. Given that $*$ is associative, what ...
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30 views

Define a matrix operation

The matrix product is defined as $$(AB)_{ij}=\sum_{m}a_{im}b_{mj}$$ What kind of the opeation is what is below? $$(A ? B)_{ij}=\sum_{m,n}a_{im}b_{nj}$$
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35 views

Proving closure

I'm curious. I have this set $$\mathbb{R}\setminus\{-1\}$$ and a binary operation defined by: $x*y = x+y + xy$. How do I prove closure of this operation? It seems obvious to me that that operation ...
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37 views

Proving Non-Associativity of a Binary Operation in $\mathbb{R}$

The operation $(*)$ is defined as $$a*b=|a-b|, \forall a,b \in \mathbb{R},$$ and I am to prove that $(*)$ is not associative in $\mathbb{R}$, that is, to prove that it is not true in general that ...
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1answer
30 views

Which of the following rules are operations on the indicated set?

Which of the following rules are operations on the indicated set? $$ a*b=a \ln (b) $$ on the set $$ {\{x \in \mathbb{R}: x>0}\} $$ I said no, because we can rewrite $a \ln (b) $ as $ \ln (b^a)$ ...
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30 views

What are other examples of complex associative operators besides, x + y +rxy, rxy, and x + y + 1/r?

I have been having fun (and frustration) in finding complex associative operators over the complex numbers. So far, I have found the 3 listed in the title (r is a constant), and also know about ...
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30 views

A question concerning binary operations and isomorphisms

Consider two nonempty sets $A$ and $B$, such that $A$ is isomorphic to $B$. Now, if $B$ is a group under some binary operation $*$, does it necessarily imply that there exists an operation $*'$, under ...
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51 views

How many numbers from $1$ to $2^n$ will have $``11"$ as substring in binary representation?

For example say, $n = 2$. So our set is $\{1, 2, 3, 4\}$ in base $10$ and $\{1, 10, 11, 100\}$ in base $2$. So Output $1$, because only one number i.e. $3$ is there such that it has $``11"$ in it. ...
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1answer
50 views

Prove by induction that every complete $k$-ary tree of depth $n$ has $(k^{n+1}–1)/(k-1)$ nodes for all integers $n\ge 0$, where $k\ge 2$.

A strictly $k$-ary tree is a $k$-ary tree (a binary tree is a $2$-ary tree) in which every node has either no children (is a leaf) or $k$ children. A complete $k$-ary tree of depth $n$ is a ...
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What am I doing wrong when multiplying binary numbers together?

This is from Discrete Mathematics and its applications I was able to get sum pretty easy. I am trying to follow this example in the book to get the product of the two binary numbers Here's my ...
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23 views

Are there alternative nonsymmetric symbols to signify nonsymmetrical operations like $-$, $|$ etc…

Commonly it's just assumed that $5-4$ means $5+(-4)$ and $7\div2 = 7\times\frac{1}{2}$, but $-$ is symmetric so I was wondering if there are there some nonsymmetric symbols like $5\rightharpoondown4 = ...
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34 views

Show that there is an identity element with respect to the operation $+$ and every subset $A$ of $D$ has an inverse with respect to $+$

Let $P_D$ be a power set of $D$. The operation $+$ is to be regarded as an operation on $P_D$. Show that there is an identity element with respect to the operation $+$ and every subset ...
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68 views

About the properties of $x * y = \frac {xy}{x + y + 1}.$

Let $x * y = \frac {xy}{x + y + 1}.$ $x * y = \frac {xy}{x + y + 1} \neq \frac {x + y + 1}{xy},$ so $*$ is not commutative. $(x * y) * z = \frac {(xy)z}{x + y + z + 2} = \frac {x(yz)}{z + y + x + ...
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55 views

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 = ...
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17 views

Associative Binary Operation from associative Binary Operation

if $\Delta$ is an associative composition(Binary Operation) on $\mathbb{E}$ and if $a\in \mathbb{E}$, then the composition $\Omega$ on $\mathbb{E}$ defined by $x\Omega y=x\Delta a\Delta y$ is ...
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114 views

Prove that the following set is a group

Prove that that the following is or is not a group. (a) The set S = $\mathbb{R}$ \ {0} with operation defined by a * b = 2ab for all a and b in S. (On the right side of the equation, the operations ...
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32 views

Binary operation $xy$, has identity, but not associativity. Is the inverse unique?

Let $S$ be a set with a binary operation $xy$ defined on it, with a neutral element, but not satisfying associativity. I want to prove that the inverse isn't necessarily unique. My attempt to answer ...
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73 views

Strange binary operation: $a\oplus b = (a-b)||(a+b)$

Motivation: A facebook post had a bunch of these as an 'intelligence' test, so I thought I would think about what this operation is. But I haven't done this Math in years! I have an operation ...
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76 views

Find number between $A$ and $B$ with maximum set bits?

Given two integers $A,B$. Find number $N$ which has maximum number of set bits in its binary form and lies between $A$ and $B$ inclusive. Is there any approach for this question. Also if there are ...
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67 views

Using XOR operation repeatedly

There are $n$ binary digits, from $A_0$ to $A_{n-1}$. Each operation consists of the following 2 steps: Each digit is replaced by the XOR addition of itself with the next digit. ...
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61 views

Is distributivity sufficient to define composition?

Function Composition has the property of distributivity: $$(f\star g)\circ h = (f\circ h)\star(g\circ h)\;\forall f,g,\star \in\{+,-,\times,\div\}$$ I was wondering if these properties uniquely ...
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32 views

How many numbers with given amount of ones in their binary form?

I was practicing for a programming competition and I got the following problem, which I was unable to solve: It is given a number N. Find the amount of x, y values, where x > N, y < N and the ...
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2answers
121 views

binary subtraction

I am trying to solve binary subtraction: $$11000_2 - 1011_2 = 1001_2$$ I know that it should be $1001_2$, when checking with answer key however I am not sure how it was calculated as I get different ...
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1answer
25 views

float vector to binary integer vector transformation preserving dot product

Is there a transformation of a set of float vectors to a set of binary integer vectors that preserves the dot product. I found conformal transformations but I'm interested in large vectors (size 300) ...
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524 views

How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
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1answer
21 views

Binary Subtraction with negative result

I want to do this little subtraction (but with bits): $1372 - 9714$ The binary code I found for $1372$ is: $00010101011100$ The binary code I found for $9714$ is: $10010111110010$ Then I added a ...
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4answers
171 views

Binary operation commutative, associative, and distributive over multiplication

Is there any binary operation that is commutative, associative, and distributive over multiplication? I asked this question in my head a while ago, and I posted it in various forums. However, having ...
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65 views

Can a smooth function on the reals form a non-commutative semigroup?

Let $f\colon \mathbb{R}^2 \to\mathbb{R} $ be a smooth function. Can there exist an algebraic structure $(\mathbb{R}, \cdot)$ such that for $x,y \in \mathbb{R}$, $x \cdot y = f(x,y)$ that is a ...
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1answer
75 views

How do I complete this “Cayley table” or binary operation table?

I have an algebraic structure $(S,\cdot)$ and $a,b,c,d \in S$ where $a,b,c,d$ are not necessarily four distinct elements. This is part of a larger problem that I am working on and based on what I ...
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53 views

Converting a boolean expression into CNF and DNF

Is there any systematic way to convert the following boolean expression (QUBO) into CNF or DNF? Here, $x_1, \ldots, x_n \in \{0, 1\}$, $a_1, \ldots, a_n \in \mathbb{Z}$ and $b_{1,1}, \ldots, b_{n,n} ...
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1answer
51 views

how to find the binary expansion of any number in the unit interval [0,1]

For each integer $n\geq 1$ and $x\in [0,1]$, define $f_n(x)=x_n$ where $x_n$ is the $n$th binary digit of x. If x is a number with two binary expansions, use the expansion that ends with infinitely ...
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1answer
14 views

Mathematical Terms for Binary Operations

I'm trying to represent binary operations on numbers in mathematically correct terminology. For example given two binary numbers: 42 : 101010 13 : 001101 I want ...
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33 views

Distributivity of $\times$ over $+$

I had a problem recently as part of class work that dealt with sets and the conditions imposed by them. This is part of a larger question that I simplified things down to. The parts I'm concerned with ...
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1answer
67 views

Association, Commutation and Identity Elements on Binary Operations?

Is the following closed, associative or commutative? f(a, b) = (a+b)/2, where a, b ∈ Z. I found that it is not closed but I am not sure how to find whether or not it is associative (I was confused ...
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1answer
31 views

Which of the following sets are closed under the binary operation $*$ defined as:$a*b=\frac{a+b}{1+ab}$

Which of the following sets are closed under the binary operation $*$ defined as: $$a*b=\frac{a+b}{1+ab}$$ $1.\{x\in \mathbb{R}:x\geq 0\}$ $2.\{x\in \mathbb{R}:|x|>1 \}$ $3.\{x\in ...
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42 views

Number of different magmas up to isomorphism

Let $(M,\circ)$ be a magma over a finite set of order $n$. I tried to count all the possible magmas up to isomorphism, but I just can't get it right. My naive approach was to count all the possible ...
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1answer
31 views

Transitive Relations on a set

I am trying to study binary relations (for myself, it's not an assignment!) I have the set $\{1,2,3,4\}$, and one of the relations in the exercise is $\{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)\}$. A ...
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1answer
88 views

How to define a taxonomy of non associative operations?

Let $A$ be a set, and let $a,b,c\in A$. Let also $\circ: A\times A\rightarrow A$ be a binary operation on $A$. We agree as usual to write $a\circ b$ to mean $\circ(a,b)$. We say that $\circ$ is ...
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33 views

Where can I find a binary calculator that can do exponentiations, roots and logarithms?

I've searched on Google, but all I found was binary calculators that can do additions, subtractions, multiplications and divisions, nothing else.