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.

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4answers
34 views

Why do parentheses make - a + when $x^2+x-x-1$ is changed to $(x^2+x)-(x+1)$?

I know that the equations are equivalent by doing the math with the same value for x, but I don't understand the rules for changing orders or operations. When it is not the first addition or ...
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1answer
35 views

Boolean expression for a problem

I want to express problems like this in boolean expression with say $XOR$, or operations etc. $HD$ = Hamming distance Say for $HD(2^4, 0000)\geq2\;$ the boolean expression is $$x1 (x2+x3+x4) + x2 (...
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1answer
80 views

Is $*$ associative?

Suppose $*$ is a binary operator on a set $A$ such that $\forall x,y\in A,$ we have $$x*(x*y)=y$$ and $$(y*x)*x=y.$$ Is $*$ associative? I can show that $*$ is commutative, because each of the ...
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1answer
24 views

Must a binary relation on a set $X$ be defined for all elements in $X$?

When defining a binary relation $R \subseteq X^2$, does there have to be a definite "true" or "false" value for a pair $(x,y) \in R$, or does it only have to be "true" to be included, and excluded ...
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1answer
66 views

Express a set as boolean function

HD = Hamming distance For a 4-bit string = x, I want to be able to express ALL other binary bit strings in a set that is a multiple of certain HD (in this example say 2) away from x AND at least that ...
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1answer
25 views

Show that if $H$ is a subgroup of the group $(G, *)$ that contains all natural numbers $k\ge4$ then $H$ contains all rational numbers $q>3$.

Consider the group $(G, *)$ with underlying set $G = (3, \infty)\subset\Bbb R\,$ and operation $$x * y = (x-3)(y-3)+3$$ I have to show that if $H$ is a subgroup of the group $(G, *)$ that contains ...
4
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0answers
36 views

Is there any name for this property of quasigroups: $x \times (y \times (z \times t)) = (x \times y) \times (t \times z)?$

Is there a name for the property $x \times (y \times (z \times t)) = (x \times y) \times (t \times z)$? Some basic facts about it I was able to figure out: It is shared by all four basic arithmetic ...
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2answers
27 views

Prove the sum and product of real numbers are continuous functions

To start with, let us study the continuity of the sum of real numbers. Precisely speaking, let us consider the function $f:\textbf{R}^{2}\to\textbf{R}$ such that $f(x,y) = x + y$. We want to prove ...
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3answers
103 views

Is there an equivalent of $\pm$ but for $\times$ and $\div$? [closed]

Is there mathematical sign that combines multiplication ($\times$) and division ($\div$) operations? Would that ever be used? For example $4({\times}{\div})2$ would give $8$ and $2$, like $4\pm 2$ ...
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1answer
17 views

program to find All numbers y , less than a number x having the bits set at places only where they are set in x.

consider for example 118 its binary representation is 1110110. I want to have all numbers that have bits set at places only at places where it is set in 118 as follows: 114->1110010 112->1110000 102->...
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Reduce and operation equation

I want to reduce this equation into (A & B) form Equation: (X & Z) * (Y & Z) Here & is bitwise AND operation (http://www.xcprod.com/titan/XCSB-DOC/binary_and.html) $*$ is ...
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0answers
50 views

Finding Bitmask which when ANDED to mask 2 numbers maximize the product of the masked numbers formed

Given two numbers A, B find M (which will be used as AND bitmask) such that (X <= M <= Y) which maximizes the product of the masked A (A & M) and masked B (B & M) ((A & M) * (B &...
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0answers
30 views

Direct decomposition of a magma onto ideals

Let's call the product of submagmas $A \cdot B$ a direct decomposition of a magma $M(\cdot)$ if: $A \cdot B = M$ ($m = a \cdot b$ for any element $m$ of $M$, where $a$ is an element of $A$ and $b$ is ...
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1answer
16 views

Find all algebras with one binary operation and the set {0,1}

I believe the only algebra is (A,*). +,-,/ don't meet the requirements. But how do I prove that there is no other? What are all the binary operations on numbers?
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4answers
29 views

Given the binary operation $x*y=x^2+4xy+y^2$ show that $a*1 \in \mathbb{N}$ has infinitely many solutions, where $a$ is irrational.

Consider the binary operation: $$x * y = x^2 + 4xy + y^2$$ defined on $\mathbb{R}$. I have to show that there are infinitely many irrational numbers $a$ such that $a * 1$ is a natural number. This ...
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2answers
30 views

About binary subtraction: how does this work?

So there's a course I'm watching online and it operates binary subtraction this way: asking y-x, where: y=0111 (or decimal 7) x=0010 (or decimal 2) And instead of using a 2's complement to change ...
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0answers
24 views

Are there any functions $f$ and $g$ such that $f(f(a, b), c) = f(a, g(b, c))$? See requirements in text.

Is there any "reversible" function $f(a, b)$ that can make successive encodings (applications of f), using different $b$'s over the results, in a single step ...
4
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1answer
51 views

Prove or disprove that, if $a*b$ has an inverse with respect to $*$, an associative binary operation with identity $e$, then so do $a$ and $b$.

This question had two preceding parts. First, to prove that the inverse of an element $a$ is unique and second, to prove the converse of this statement. I.e. if $a$ and $b$ have inverses, then so does ...
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1answer
26 views

Can “ multiplicative inverse of $\frac nm = \frac mn$” be obtained directly from the definition of multiplicative inverse: “ inverse of $n=\frac 1n$”?

Note : from, if I dare say, a semantic point of view, " multiplicative inverse of a number $N$ " is clearly defined as " a number $M$ such that $N\times M = M\times N = 1$" ( the multiplicative ...
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0answers
20 views

Extend hyperoperation series to index in $\mathbb{R}$ [duplicate]

The hyperoperations are a series of binary operators that are created via iteration. The zeroth member in this series ("operation 0") is a simple increment , $a' = a+1$. Incrementing the number $x$ $...
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10 views

induced binary operation on empty subset

Let a binary operation * be defined on a nonempty set S. Is it possible to define the induced binary operation on the empty subset of S? I think it's true since not having any elements automatically ...
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0answers
21 views

Is the Hamming distance sensitive to sequence of bits?

Two binary strings ($110$) and ($100$) have a hamming distance of $1$, similarly ($110$) and ($111$) also have hamming distance $1$. Hamming distance is not sensitive to the sequence of bits. Is there ...
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1answer
60 views

is the distributive property for $2+ (x-1)$?

Does the distributive property hold for a formula having as form : $( a+(x-b))$? For example, suppose $2+ (x-1).$ Applying the distributive property would be like $ (2 + x) - (2 + 1) $ is this ...
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1answer
148 views

Find the number of solutions for $\underbrace{x*x*\ldots*x}_{x\text{ 10 times}}=\frac{1}{10}$ with a given binary operation.

On the interval $(-1, 1)$, consider the binary operation $$x*y=\dfrac{2xy+3(x+y)+2}{3xy+2(x+y)+3}$$ with $x, y \in (-1, 1)$. I have to find the number of solutions for the equation: $$\underbrace{...
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3answers
174 views

Models of a certain (weird) equational theory

Consider the following (single-sorted) equational/algebraic theory with one binary operation symbol $\ast$ whose axioms are as follows: $$(x \ast x) \ast (x \ast x) = x$$ $$(x \ast y) \ast (x \ast y) =...
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1answer
73 views

Is an associative binary operation with trivial squares necessarily commutative?

Take a set $S$ and an associative binary operation $*:S \times S \rightarrow S$ such that there exists an element $e$ such that $x * x = e$ for any $x \in S$. Can we conclude that the operation is ...
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1answer
40 views

Prove or disprove that if a and b have inverses with respect to ∗, then so does a∗b (where * is an associative binary operation with an identity e)?

This is part 2 of a question. Part 1 I was proving that the inverse of an element $a$ is unique with respect to $\cdot$ and I think I solved that one. Part 3 of the question is the opposite of part 2:...
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3answers
67 views

How to deduce the “divide by a fraction” formula from the definition of division

From the comments I got, my question amonts to : can the " inverse of inverse" law be derived from the definition of division. Division is defined as : $\dfrac AB = A.\dfrac1B$, that is, " ...
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1answer
21 views

How many bits are needed to encode the following?

1) Students in this section (there are 57) 2) Students in this class (there are 117) 3) The range from 327 to 234,341 inclusive. 4) The range from 0 to 1,024 inclusive. 1,2,and 4 are trivial (at ...
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1answer
41 views

Prove that if a set of whole numbers is closed for subtraction, it is closed for addition. [closed]

The set is a subset of $\mathbb{Z}$. So far, I only know that for $x, y \in A : (x-y) + y \in A$, But I don't know how to prove that $(x + y)$ is also in $A$ based on this? Any direction would be ...
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1answer
30 views

Binary operation on the set of first n positive integers

Let $S$ be the set of the first $n$ positive integers. Suppose we have a binary operation @ that takes $a, b \in S$ to some $a @ b \in S$. Given that: 1 @ x = x @ 1 = x x @ (y @ z) = (x @ y) @ (x @ ...
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2answers
35 views

How many binary operations with a zero element can be defined on a set $M$ with $n$ elements in it?

Let's say I have a set $M$ of $n$ elements, with $n \in \mathbb{N}^*$. I have to answer the following question: How many binary operations with a zero element can be defined on $M$? I know that $*$...
23
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4answers
1k views

Does symbol “$+$” denote an operation in the notation of a complex number: “$a+ib$”? In case it does, which operation does “$+$” denote?

This is a beginner's question. A complex number is an element of R², that is an ordered pair (a,b) , the numbers a and b being elements of R. A complex number can be written : a + ib . I know ...
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2answers
34 views

Representing a sentence using propositional logic

I am confused regarding a propositional logic representation of a sentence. Please note that this sentence is not realistic: "A person who is male (M) is smart (S) if he is tall (T), but otherwise ...
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1answer
20 views

Proving pair consisting of a set and binary operation is a group and whether it is Abelian.

This question is in relation to Group Theory. I am trying to determine which of the following pairs consisting of a set and a binary operation ($G$, *), is a group. And which are Abelian groups. $1....
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1answer
58 views

What is the identity element of 4

If $*$ is a binary operation taking the greater of two distinct numbers, construct a table for the operation on the set $S=\{1,2,3,4,5\}$. What is the identity element of 4? Is the operation ...
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2answers
49 views

let s be a binary set with a binary operation * and identity element e. let $a^{-1}$ be an inverse then $a^{-1}$ is the only element in S

Question: Let 𝑆 be a set with a binary operation ∗ and identity element 𝑒. Let $𝑎^{−1}$ ∈ 𝑆 be an inverse for some element 𝑎 ∈ 𝑆. Then $𝑎^{−1}$ is the only such element of 𝑆 with this ...
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2answers
46 views

Prove that the binary operation a*b=a does not contradict the theorem 2.1.8

Suppose we have the binary operation ∗ defined on ℝ by 𝑎 ∗ 𝑏 = 𝑎. Then every 𝑎 ∗ 𝑏 = 𝑎 for every 𝑎 ∈ ℝ. For example, 𝑎 ∗ 2 = 𝑎 and 𝑎 ∗ 7 = 𝑎. Why doesn’t this contradict Theorem 2.1.8 ...
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1answer
52 views

Prove that if a set 𝑆 has an identity element, 𝑒, under ∗, then 𝑒 is always its own inverse

A binary operation * on a set is a mapping 𝑓:𝑆 × 𝑆 → 𝑆 that takes any two elements 𝑎,𝑏 ∈ 𝑆 to exactly one element 𝑓(𝑎,𝑏) ∈ 𝑆, which will be denoted 𝑎 ∗ 𝑏. We know that by definition ...
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3answers
167 views

What exactly is an “induced operation”?

Examples of places where I see this used: Let * be a binary operation in S and let H be a subset of S... the binary operation on H given by restricting * to H is the induced operation of * on H ...
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1answer
54 views

Why does shifting right on a two's complement binary number divide it by 2?

I understand how logical right shift works. Given an unsigned binary number $n = a_{n-1}a_{n-2}...a_0$, digit $a_k$ contributes $a_k\times 2^k$ to the value of the $n$. after applying a logical right ...
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0answers
38 views

Finding a vector space or group structure on a set of complex 4-tuples

Consider the set of 4-tuples of complex numbers $(x_1,x_2,y_1,y_2)$ for which $$\Im\left( -\frac{-y_1 x_1-y_2 x_2}{x_1^2+x_2^2} \right)=0 \tag{1}$$ and $$\Im\left(\frac{i (y_1 x_2-y_2 x_1)}{x_1^2+...
0
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1answer
21 views

Binary Addition - Carry one when converting from negative to positive base-2 number?

I am doing a binary addition problem with two base-16 values and the one’s complement system. The answer I got on my problem is off by just 1, and I am wondering why that may be. Whenever you are ...
0
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1answer
35 views

does the binary operation have an inverse element and or identity element

$a*b = \sqrt{ab} \space a,b \in \mathbb{R}$ It is easy enough to show it is communitive and not associative: $a*b=\sqrt{ab}=\sqrt{ba}=b*a$ thus commutative $(a*b)*c= \sqrt{ab\sqrt{c}} \neq\sqrt{a\...
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3answers
51 views

Binary Multiplication Result Size Length

I am trying to understand why multiplying two binary numbers of size L gives a resulting sized number of O(2L), Is it becuase the 'worst case'/maximum size possible is 2L ? Which is when the two ...
0
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1answer
62 views

Prove an operation is associative

Given multiplication on natural number s.t. a * b = r where r is the remainder when the product of ab is divided by divisor n. I need to show that the operation is associative. Based on the ...
0
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0answers
30 views

Find an identity of an operation

Given x * y = |x - y|, we need to find the identity of the operation. Based on the definition of the operation, x * e = e * x = x. Thus, let x * e = x so we get |x - e| = x. We have e = 0 or 2x. ...
0
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1answer
31 views

Is this a Cartesian product set or just a binary operation?

If we consider this equation $$S+J=11$$ where $S$ and $J$ are Sam and Jane and their combined age is $11$, and I don't allow for one to be $11$ while the other is $0$ years of age, then I have a ...
0
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2answers
63 views

Binary number in compact form

Can we represent a binary number in a compact binary form. For example 4294967295 (decimal) -> 11111111111111111111111111111111 (binary, 32 bits) Can we ...
0
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1answer
55 views

Boolean function expression

I have an interesting problem. How do I express the following as a boolean function? HD ($2^4$, 1100) >=2 HD = Hamming distance $2^4$= {0000,0001,0010.....all 16 binary values} The answer ...

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