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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How to express the min operator as a binary operator

I'd like to use the $\min$ operator as a binary operator that returns the lowest of two given numbers. I'm not sure if this is the correct use of it, or if I should use something else. $\text{Given ...
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27 views

Visual or faster way of multiplication of two $3$ digit numbers

There is a (a bit lengthy , not useful, but visual) way to multiply numbers. Case-1 If we want to multiply two one digit numbers, say $4\times 5$, then we can see the answer as number of ...
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30 views

Algebraic structures and axiomatic systems

In one textbook appears the following sentence: An algebraic structure is a nonempty set $M$ together with one or more operations (i.e. a function $*:M\times M\rightarrow M$) which satisfy some ...
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50 views

How do we deal with units when using the modulo operation?

I'm wondering how I should deal with units when I do a modulo operation. What is considered legal and what is not. When I have two numbers that have units such as 13cm and 3cm, I can multiply them: ...
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43 views

External operation: binary and unary perhaps???

Consider the following examples from which some definitions are derived: Let us take an element from the set R of real numbers (say, the number 8) and another from the set L of lengths (say, 4m). ...
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28 views

Finding binary operations on connected graphs

If $G = (V,E)$ is a connected graph with $||V|| \geq 2$ , $W(G)$ being the set of all paths in $G$. How do you find a binary operation $ +$ on $W(G)$ such that $\langle W(G),+\rangle$ is an algebra ...
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1answer
70 views

Finding a binary operation on $\{1, \dots, n\}$ so that each $k$ has exactly $k - 1$ left inverses

What is an example of a binary operation on the set $\{1, \dots, n\}$ so that each element $k \in \{1, \dots, n\}$ has respectively $k-1$ left inverses? I have been trying various combinations ...
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15 views

Function mapping notation for a binary operation on a set

I found the following definition of a binary operation on a set from here: A binary operation $*$ on a set $S$ is a map $\ast:S\times S \rightarrow S$ My question is, if I define an operation ...
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1answer
65 views

Binary operations in an algebra

Is there a binary operation ° for the algebra <{1,...,n},°> such that for each $k \in \{1,...,n\}$ there are exactly $k-1$ ...
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23 views

Semigroup where the Binary Operation is not Associative.

I am working on my functional composition, which has the associative property, to show if a given pair is a semigroup or not. I believe all Semigroups have to have a binary operation that is ...
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4answers
47 views

Binary Operations $x,y \in S$

Trying to understand binary operations, but seriously confused. I was looking at all the videos on youtube, forums, but I think I must be missing something. I have a set $$S = \{a, b, c, d, e \}$$ ...
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109 views

How do we add numbers?

How do we compute sums in general? How can we tell the result of the operation $A+B$? Even when we talk about very basic numbers like $\Bbb{N}$ I find it hard to understand the algorithm we use to ...
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5answers
199 views

Which are the operations used in mathematics? [closed]

Everyone knows +,-,x,:,^. But I would really like to know which other operations exist, and what they do.
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1answer
38 views

Does this notion of “weak” isomorphism exist in literature?

Let $(M,\circ)$ and $(N,\ast)$ be two magmas. I'd like to relax the notion of isomorphism by defining a notion of "weak" isomorphism in the following way: $M$ and $N$ are "weakly" isomorphic if there ...
2
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2answers
41 views

Computing a certain $2014$-fold product using a particular associative binary operation $\ast$

$$x*y = 3xy - 3x - 3y + 4$$ We know that $*$ is associative and has neutral element, $e$. Find $$\frac{1}{1017}*\frac{2}{1017}*\cdots *\frac{2014}{1017}.$$ I did find that ...
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16 views

Binary arithmetic with unsigned numbers

I am struggling with performing binary math with unsigned numbers. I know I am supposed to take the 2s complement (flip the bits and add 1) of the subtrahend before I subtract, but for some reason my ...
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2answers
33 views

How do I interpret this operation?

This question has to do with operations and exploring their characteristics. I have just learned how to extract info from an operations table (what is the identity, inverse, etc.), but this question ...
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33 views

Binary division using polynomial

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
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1answer
24 views

Explanation of Distance of binary vectors formula

So, here's once again this article from topcoder about combinatorics. After the article successfully describes what theory it will use: Combinations/Permutations, it goes into an application for it, ...
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1answer
22 views

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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1answer
46 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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1answer
29 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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3answers
28 views

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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1answer
38 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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1answer
34 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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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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40 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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32 views

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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1answer
65 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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18 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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3answers
52 views

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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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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43 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
34 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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1answer
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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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2answers
36 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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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
80 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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49 views

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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27 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 = ...
0
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1answer
45 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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69 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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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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1answer
18 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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213 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 ...
0
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2answers
48 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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1answer
84 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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1answer
82 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 ...
0
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73 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. ...