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
1answer
24 views

How do you prove the left and right side of an identity of a set are equal?

I'm having some trouble understanding sets w/ associative binary operations. Say I have a set "S" w/ the associative binary operation SxS -> S. If 'L' is a left identity of S and 'R' is a right ...
-3
votes
0answers
13 views

Conditions necessary for the existence of inverse [on hold]

What are the 3 Conditions necessary for the existence of inverse of a given element under a binary operation on a set S
1
vote
2answers
45 views

Suppose that $\cdot$ is associative and has an identity element. Show that an element $g \in G$ has at most one inverse

Let $(G,\cdot)$ be a group with $e$ its neutral element. For an element $g\in G$, there exists one inverse element in $G$, denoted by $g^{−1}$, such that $g\cdot g^{−1}=g^{−1}\cdot g=e$. Can this be ...
-2
votes
0answers
13 views

Forming a Cayley table [closed]

How to form a Cayley table using $*$ as a binary operation on $P(A)$, where $A=\{1,2,3\}$. And the solution of main diagonal in the Cayley table.
3
votes
0answers
23 views

Why calculating XOR of consecutive values can be simplified?

I was trying to calculate integer xor of 0..n. I named the function xored(n). Note that in examples below ^ does not mean power but integer xor (like in C or Java language) So, xored(0) = 0, ...
3
votes
1answer
29 views

How to solve (0.1 - 10.3 + 5.132)/12.8 and round off correctly?

I've recently learnt the rules about rounding off when adding/subtracting and when multiplying/dividing. I know that when you add/subtract, the number of decimal places in the result should equal the ...
-4
votes
1answer
28 views

Binary word addition; error pattern

If a word $a = a_1,a_2,...,a_n$ is sent is sent (this is in regards to coding/IT/etc.-I'm trying not to include any extraneous information) and a word $b= b_1,b_2,...,b_n$ is received (where the ...
0
votes
1answer
23 views

Identity element of word addition

I realize this is rather an arbitrary question, but it's important to me, that I understand it and get it right, and I'm not finding the answer anywhere else. I'm working through "A Book of Abstract ...
0
votes
0answers
22 views

KoreK's ChopChop Attack (Inverse Arbaugh Inductive Attack)

I would like to know how KoreK's "ChopChop" attack on the WEP protocol works on a basic mathematical level. His original source code provides little clue as it uses precomputed values, the origins of ...
3
votes
0answers
76 views

Is the operation $x*y=xy/(x+y+1)$ associative or not?

This questions is from Charles C. Pinter's "A Book Of Abstract Algebra, Second Edition". This is the 7th question of Chapter 2 on page 24. In the selected answers section it says it is not associative ...
1
vote
1answer
29 views

Operation table for A+B where + denotes the operation of symmetric difference

If someone could please verify if my operation table in the picture below is correct it'd be much appreciated. The task was given: $P_D=\{A: A \subset D\}$ and $D$ is a $3$-element set $D=\{a, ...
3
votes
1answer
22 views

Convolution can smooth an input function, is there an operation which bunches it up?

An easy to remember description of what the convolution of two functions is, is to say that one is a weight function and the result is a weighted average of the other function. The canonical example ...
1
vote
1answer
59 views

A proof that the operation of concatenation has an identity element

I'm relatively new to proofing and am wondering if this is an acceptable proof. The book for anyone who would like to reference it is "A Book of Abstract Algebra" by Charles Pinter. It is problem ...
2
votes
1answer
57 views

Proof of associativity for concatenation operation

I recently took a mathematical proof class and am beginning to teach myself abstract algebra. I'm fairly new to proofing however, and am not very confident in how I do it. Also, I'm new to this site ...
2
votes
1answer
94 views

Operation $\circ$ on $\mathbb{R}: x\circ y=x\sqrt{1+y^2}+y\sqrt{1+x^2}$

I have the following operation $\circ$ on $\mathbb{R}: x\circ y=x\sqrt{1+y^2}+y\sqrt{1+x^2}, \forall x, y\in \mathbb{R}$. My question: are the groups $(\mathbb{R},\circ)$ and $(\mathbb{R},+)$ ...
2
votes
1answer
50 views

Integer programming: if a or b then a, b, and c

I'm writing a mixed integer programming (MIP) constraint where my $\color{blue}{\texttt{binary variables}}$ are $a, b,$ and $c$ to meet the following condition: $$ (a \lor b) \to (a \land b \land c)$$ ...
5
votes
2answers
183 views

Commutative binary operations on $\Bbb C$ that distribute over both multiplication and addition

Does there exist a non-trivial commutative binary operation on $\Bbb C$ that distributes over both multiplication and addition? In other words, if our operation is denoted by $\odot$, then I want the ...
0
votes
1answer
75 views

Of the 16 binary operations on a two element set, which ones are commutative, associative, have an identity element, and have inverse?

If you you 16 binary operations $$(a*a)=a$$ $$(a*b)=a$$ $$(b*a)=a$$ $$(b*b)=a$$ $$(a*a)=a$$ $$(a*b)=b$$ $$(b*a)=a$$ $$(b*b)=b$$ $$(a*a)=b$$ $$(a*b)=a$$ $$(b*a)=b$$ $$(b*b)=a$$ $$(a*a)=b$$ ...
2
votes
1answer
32 views

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 ...
0
votes
0answers
34 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 ...
3
votes
1answer
33 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 ...
1
vote
2answers
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: ...
1
vote
1answer
44 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). ...
3
votes
0answers
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 ...
2
votes
1answer
71 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 ...
0
votes
2answers
23 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 ...
1
vote
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 $k-1$ ...
0
votes
2answers
27 views

Semigroup where the Binary Operation is not Associative. [closed]

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
48 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
113 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
206 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
41 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
22 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
34 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
34 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
25 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
24 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
65 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
36 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
29 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
42 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
36 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
101 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
61 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
33 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
79 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
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 $i=1,...,T$ ...
1
vote
3answers
54 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? $$(A ? B)_{ij}=\sum_{m,n}a_{im}b_{nj}$$