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

learn more… | top users | synonyms

0
votes
2answers
10 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 ...
0
votes
0answers
24 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 $k-1$ ...
0
votes
2answers
18 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 ...
0
votes
4answers
45 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 $$S = \{a, b, c, d, e \}$$ ...
-1
votes
2answers
105 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 ...
4
votes
5answers
192 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.
2
votes
1answer
36 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
votes
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 ...
0
votes
0answers
14 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 ...
0
votes
2answers
30 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 ...
1
vote
0answers
32 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 ...
0
votes
1answer
21 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, ...
0
votes
1answer
20 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 ...
1
vote
1answer
37 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 ...
1
vote
1answer
27 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 ...
0
votes
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 ...
3
votes
1answer
35 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 $X$ ...
2
votes
1answer
33 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 ...
4
votes
2answers
78 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 ...
0
votes
1answer
31 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 ...
2
votes
0answers
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 $1's$ ...
2
votes
1answer
54 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, ...
0
votes
0answers
16 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 $i=1,...,T$ ...
1
vote
3answers
50 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 ...
0
votes
1answer
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? $$(A ? B)_{ij}=\sum_{m,n}a_{im}b_{nj}$$
1
vote
1answer
42 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 ...
3
votes
0answers
41 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 ...
1
vote
1answer
33 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)$ ...
1
vote
1answer
42 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 ...
1
vote
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 ...
5
votes
1answer
52 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. ...
0
votes
1answer
69 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 ...
1
vote
2answers
48 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 ...
1
vote
0answers
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
votes
1answer
43 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 ...
0
votes
1answer
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 + ...
1
vote
2answers
57 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 = ...
0
votes
1answer
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 ...
1
vote
3answers
178 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
votes
2answers
44 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 ...
0
votes
1answer
77 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 ...
0
votes
1answer
78 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
votes
1answer
70 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. ...
4
votes
1answer
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 ...
1
vote
2answers
37 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 ...
0
votes
2answers
125 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 ...
2
votes
1answer
28 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) ...
1
vote
4answers
1k 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 ...
0
votes
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 ...
6
votes
4answers
188 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 ...