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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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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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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25 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.
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22 views

Express a binary operation in decimal

Is there a way to represent binary operation in decimal. What I mean with this is for example a set of decimal operators that would give the same result as a x>>n a ror(x), etc. So far the only thing ...
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39 views

Absolute Value Equivalence relation inequality Question

I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me? For $a, b ...
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29 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
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42 views

Decimal binary sequences that cannot be greater than $1$

Consider the family of sequences of the form $.012\ldots n$ for any natural number $n$. So, the sequences in this family are: $.01, .012, .0123, .01234,$ etc. Now consider to manipulate each ...
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Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
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47 views

Does the binary operation $m ⋆ n = m^n$ on $\mathbb N$ have a neutral element?

Does the binary operation $\,m ⋆ n = m^n\,$ on $\,\mathbb N\,$ have a neutral element? I said yes, and it is $\,e=1\,$ because $\,m ⋆ e = m^e = m^1 = m,\;$ but apparently that is wrong.
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Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
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How to compute associative binary operation on a finite set based on partial information?

I am working on a problem, and I must be staring at the answer without seeing it since it's among the introductory problems in my abstract algebra textbook. We're told that an associative binary ...
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1answer
27 views

Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
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1answer
28 views

Associative law with negative sign

We know that $a+b+c$ is meaningful for, say, the group $\left( \mathbb{Z}% ,+\right) $. Since for any $b,c$, we must have $b^{-1},c^{-1}$, therefore $% a+b^{-1}+c^{-1}$ has to be meaningful, too, but ...
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1answer
31 views

How do I prove (scalar1 + vector1) * scalar2 is not equal to scalar1 * scalar2 + scalar2 * vector1?

I am taking a Linear Algebra course and have been stumped on a homework question for a few hours. How do I prove for two scalars, $c_1$ and $c_2$, and a vector $v$: $(c_1 + v)c_2 \neq c_1c_2 + c_2v$ ...
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2answers
63 views

Is this binary operation a group?

Let $Y=\left\{(a,b)\in\mathbb{R}\times\mathbb{R}\ |\ a\ne 0\right\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)∗(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$. I already did the ...
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how to prove this group of the binary operation

Let $Y=\{(a,b)∈ \Bbb R\times \Bbb R∣ a≠0\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)*(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$. I already do the proof of ∗ is an operation on ...
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46 views

How to determine if a set is closed under some operation?

Is the set $\{-2,0,2\}$ closed under addition? And why? Specifically, when determining if a set is closed under an operation do you apply the operation to the each number and itself? For ...
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1answer
34 views

How to prove that: If two binary operations are anti-isomorphic and one of them is associative then the second one also will be associative?

We know what is called an anti-isomorphic operation on a set S. it is just a one two one $ g $ function mapping from $S$ to $S$. $ g: S \rightarrow S$. and it satisfy this condition $ g(xy)= ...
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Algebraic Structure: Are Set Operations Considered Binary Operations?

I'm currently trying to understand the "hierarchy" of sets / algebraic structures, e.g. things like groups, rings, fields, modules, algebra, vector spaces which I mostly understand, but especially the ...
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In how many ways can we multiply three numbers? [closed]

Let $a,b,c \in \mathbb{Z^+}$. In how many ways can we multiply this three numbers? At any rate, I'm very grateful for your help!
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59 views

Clarify Cartesian Products and Binary Operations

So tell me if I'm saying this write. A Cartesian Product is a function f:X x Y --> Z , where some unknown structural operation on the sets X and Y produces a set Z as its codomain, and Z is a set of ...
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For an associative binary operation with identity, the set of invertible elements forms a group

Let $S$ be a set, and $*$ an associative binary operation on $S$. Suppose there is an element $e\in S$ such that ($1$) $e*x=x$ and $x*e=x$ for all $x\in S$. (a) Prove that there is a unique element ...
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How to create a new binary operation on a same set?

I was studying binary operation on a set. Then the following question came to mind. I tried to find an answer. also searched in website but could not get any satisfactory answer. the question is: is ...
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35 views

Associative Numbers

Yesterday my friend wrote a number on a paper. He then added the number of ones in the binary representation of the number to that number and formed a new number. He kept doing the process infinitely. ...
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92 views

Explaining multiplication of fractions

The best way I've been able to describe multiplication is as $$ a\times b = \sum^a_{i=1} b$$ But my definition does not account for things such as $2.99792458\times8.987551787$ and ...
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3answers
98 views

What is a bit-shifting standard C function for calculating $f(x) = \frac{(2^{16}- 1)}{(2^{32} - 1)}\cdot x$

I need to take 32-bit unsigned integers and scale them to 16-bit unsigned integers "evenly" so that $0 \mapsto 0$ and 0xFFFFFFFF $\mapsto$ ...
4
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1answer
41 views

Is the number of associative $n$-ary algebraic operations on a finite set with 2 cardinality always 8?

We know that if $n = 2$ then the operation is called a binary operation. $ \circ $ on set $X$ is a function $\circ : X \times X \rightarrow X$. And the number of all associative binary operation on a ...
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29 views

Reducing Several Identities to One Identity

One class of algebraic structures that are typically studied are those given by a set $X$ and a set of $n$-ary operations defined on $X$ for each $n\in \mathbb{N}$. Perhaps most studied are those ...
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Multiplication by scalar axioms for an abelian group.

There is an R vector space where $k ⊙ x := x^k$ , $∀x, y ∈ V, k ∈ R$, I showed that it was abelian. I wanted to show scalar multiplication by using the axioms. $α ⊙ (x ⊕ y) = α ⊙ (xy) = (xy)^α = ...
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Prove that the group $(A,+, ◦) $ is a non-commutative ring

• $A × A → A, (f, g) → f + g$, where $(f + g)(x) = f(x) + g(x)$ for all $x ∈ K$ • $A × A → A, (f, g) → f ◦ g$ where $(f ◦ g)(x) = f(g(x))$ for all $x ∈ K$ Show that $(A,+,◦)$ is a non commutative ...
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What is subtraction?

Let $a, b \in \mathbf{R}$. It is an elementary fact that addition is a commutative binary operation on the reals, that is, $a + b \in \mathbf{R}$ and $a + b = b + a$. With the exception of ...
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1answer
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Restricting Binary Operator $*$ To A Subset

I am currently reading Dummit and Foote's Abstract Algebra, and am having a little confusion over the following excerpt: Suppose that $*$ is a binary operation on a set $G$ and $H$ is a subset of ...
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How many associative ternary operations there are on a finite set?

We know that algebraic operation is a function $f:\underbrace{\left ( X\times X\times \cdots \times X\right )}_{t\ \text{times}}\rightarrow{X}$ If $X$ is a set and and cardinality is $|X|=n$ then ...
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How is addition different than multiplication?

Is there a fundamental difference in the things we call multiplication and those we call addition? In a field, both binary operations obey exactly the same rules (commutativity, associativity, ...
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Algebraically, What Does $\Bbb R$ get us?

In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$? Say we start with $\Bbb N$: ...
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Proving $K$ is a group

Now I have proved certain things are a group before, and I know that it requires: 1)Associativity 2)Inverse 3)Identity But here I have such a strange thing that I wanted to clarify that I am doing ...
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4answers
78 views

Show that * is associative

Could you show me how to prove the following to be associative? Please take me through the process step by step. $$a*b=a+b+2ab$$ Where $*$ is a binary operation and $a$ and $b$ are real numbers. I ...
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Nonempty, associative, and closed under inverses but not a group

Given an example of a set $G$ and an operation $*$ on $G$ such that $*$ is not a binary operation on $G$ but associative, identity and inverses properties hold? Basically, try to find an example to ...
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3answers
52 views

are there meaningful binary operations on the set of Catalan objects?

Is it possible to come up with some kind of meaningful closed binary operation $\star$ on sets $C_n$ of Catalan objects? By Catalan objects I mean objects that correspond to a given Catalan number. ...
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1answer
57 views

Equivalence Relation using Binary Operations.

Question: Let ∗ be a binary operation on a set A. Assume that ∗ is associative with identity. Let R be the relation on A defined on elements a,b ∈ R as follows: aRb if there exists an invertible ...
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197 views

How do I add multiple binary numbers without using a partial sum?

I know how to add binary numbers but what I normally do is add the first 2 binary numbers and then add the 3rd one to their sum. It is really slow. $$ 111_2 + 111_2 + 111_2 + 111_2 $$ Here is ...
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Efficient way to compute the binomial using $(2^k+1)^{k+1}$

The following web page: "http://introcs.cs.princeton.edu/java/78crypto/" (at Exercise 28) effectively says that: "Pascal's triangle. One way to compute the $n$-th row of Pascal's triangle (for $n ...
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Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$

When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that $$a*(b*c)=(a\cdot b)*c$$ If $*$ is associative then $\cdot=*$ even if I'm not sure ...
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Proving that operations give equal results given equal inputs

I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as ...
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1answer
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Computing Number of Possibilities

I'm asking this question in the context of a program I'm trying to write, but the problem I'm having seemed to be more of a mathematical one. (Also, I'm not quite sure what tags should be applied to ...
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1answer
31 views

Binary operation on empty set?

Can we techniclly declare a binary operation on an empty set? Since binary operation does an action on some objects (which empty set dot have)... Thanks.
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Set composed of operations on a Subset

Let there be a set P, and a set K such that $P\subset K$. Let there be 2 binary operations closed on K written $+$ and $\times$. Is there any way to define K as having only elements composed of ...
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“Self invertible” group

Let there be an Abelian group with a binary operation $\ast$ on a set $S$. Let such a group respect the following propriety: $$ (X\ast Y)\ast Y = X$$ For any $X$ and $Y$ in $S$. I realize that by ...
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769 views

Convert from base 10 to base 5

I am having a problem converting 727(base 10) to base 5. What is the algorithm to do it? I am getting the same number when doing so: $7*10^2 + 2*10^1+7*10^0 = 727$, nothing changes. Help me figure it ...
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How do I show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab?

I really need help with this question. I am required to show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab, and I'm not quite sure how. Any work/help is ...