What is associative: Definition and 51 Discussions

In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations:

Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations".
Associativity is not the same as commutativity, which addresses whether the order of two operands affects the result. For example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation. However, operations such as function composition and matrix multiplication are associative, but (generally) not commutative.
Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative.
However, many important and interesting operations are non-associative; some examples include subtraction, exponentiation, and the vector cross product. In contrast to the theoretical properties of real numbers, the addition of floating point numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.

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  1. H

    A About universal enveloping algebra

    Please, I have a question about universal enveloping algebra: Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in...
  2. P

    I Group like operations that are not associative

    A group can be defined by the following three properties. (Source: wikipedia) Is there any example of an operation that fails the associativity test, but meets the other two tests? I'll refer to this hypothetical entity as an almost-group for the purposes of this post lacking any knowledge...
  3. karush

    MHB T31 vector subtraction is not commutative and not associative.

    Prove, by giving counterexamples, that vector subtraction is not commutative and not associative. ok I read all I could on trying to understand this but seem to not see something simple if we have the example of $u=\begin{bmatrix}2\\-3\\4\\2\end{bmatrix}...
  4. C

    B How to prove the associative law of symmetric difference?

    I'm trying to prove the associative law of symmetric difference (AΔ(BΔc) = (AΔB)ΔC ) with other relations of sets. A naive way is to compare the truth table of two sides. However, I think the symmetric difference is not a basic one, it is constructed form other relations, that is AΔB =...
  5. TheMercury79

    Is the Cube of matrix associative?

    But I actually don't get the same matrix. What I get is the transpose of the other when I change the order i.e when I do [A]^2[A] I get the transpose of [A][A]^2 and vice versa What I'm trying to do is find the cube of the expectation value of x in the harmonic oscillator in matrix form. We're...
  6. S

    B Commutative & Associative property of negative numbers

    Commutative property of addition. If a & b are integers then, a+b = b+a 2+3 = 3+2 5. Does not work for subtraction. 2-3 = -1 3-2= 1 Having said that, what about the special case with negative numbers (when we also move their respective signs) -5 + 7 = 2 & 7 + (-5) = 2. 15 -7 = 8 & -7 + 15...
  7. Specter

    Proving the associative property of vector addition

    Homework Statement Give an example of the associative property of vector addition using vectors in Cartesion form. Homework Equations (u+v)+w=u+(v+w) The Attempt at a Solution I can't figure out how to get the arrow on top of my work so I wrote it without it. I'm somewhat confused on why I...
  8. C

    MHB Proving matrix group under addition for associative axiom

    Dear Everyone, I have some feeling some uncertainty proving one of the axioms for a group. Here is the proof to show this is a group: Let the set T be defined as a set of 2x2 square matrices with coefficients of integral values and all the entries are the same. We want to show that T is an...
  9. karush

    MHB 412.x4 Determine which of the folllowing operations are associative

    Determine which of the folllowing operations are Associative, Identity, or Inverse $\textit{Additive on $\mathbb{Z}$}$ $\textit{Subtraction on $\mathbb{N}$}$ $\textit{Division on $\mathbb{R}$}$ $\textit{Division on $\mathbb{Z}/\{0\}$}$ $\textit{Composition on $D_4$}$ $\textit{Composition on the...
  10. Mr Davis 97

    I How can we use induction to prove the generalized associative law?

    I am trying to prove the generalized associative law with induction, but am being tripped up by one aspect. I am reading a solution and it says for the induction step argue that any bracketing of the product ##a_1 \cdot a_2 \cdot \cdots a_n## must break into two subproducts ##(a_1 \cdot \cdots...
  11. C

    I Is a commutative A-algebra algebraic over A associative?

    Let ##A## be a ring and ##B## be a commutative algebra over A· Suppose that ##B## is generated by algebraic elements ##\beta\in B## over ##A##, meaning that ##\beta## fulfils a relation of the form ##P(\beta)=0##, with ##P\in A[X]##. Is ##B## necessarily associative ? NOTE: As usual...
  12. D

    MHB My First Algebra Question: Associative Property

    I am reading through a introductory algebra textbook and refreshing memory on this topic. In the matter of the Associative Properties, it is giving me the expression 3 + (8+x). (I'm skipping the graphs section because I'm leaving that for my tutor Saturday. But I'm moving to algebra because I...
  13. Math Amateur

    MHB Tensor Products and Associative Algebras

    I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...
  14. C

    Prove that ordinal arithmetic is associative

    Homework Statement Let a, b, c be ordinals. Prove that a+(b+c)=(a+b)+c Homework EquationsThe Attempt at a Solution I looked at a set theory book by Jech and he says Prove by induction on c. Should I look at the case where its true for c+1[/B]
  15. G

    How do i show if this is associative or not?

    Homework Statement (x*y)=x+2y+4 Homework EquationsThe Attempt at a Solution first i did this but I'm not sure if it is correct (x*y)*z=x+2y+4*z=x+2y+4+z+1 x*(y*z)=x*y+2z+4=x+y+2z+4+1
  16. O

    Is multiplication associative in physics?

    Work = Force X Distance. ? = Distance X Force How do you make sense of the second equation?
  17. H

    Is Addition Associative? A Proof Using Induction

    Homework Statement For any natural numbers a,b,c, we have (a+b) + c = a + (b + c) Homework Equations Definition 2.2.1 Let m be a natural number. To add zero to m, we define 0 + m := m. Now suppose inductively that we have defined how to add n to m. Them we can add n++ to m by defining (n++) +...
  18. B

    Does Associative Property Apply to Subtraction and Division Too?

    I show that the assoc. property applies to addition and multiplication in my book: (a+b)+c = a+(b+c) (ab)c = a(bc) But what about subtraction and division?
  19. 1

    Is LCM Associative? Tackling the Proof

    Homework Statement I need to prove that the least common multiple operation is associative. Homework Equations The Attempt at a Solution Pages of crappy algebra trying to use the fact that LCM(a,b) = |ab|/gcd(a,b) I hate to be "that guy" that doesn't post much of an attempt...
  20. L

    Klein 4-group 2 Associative property

    Homework Statement How to prove in the easiest way that Klein 4-group is associative. Homework Equations Four elements ##a^2=b^2=c^2=e^2=e##. The Attempt at a Solution If that is group with four elements, how many types of ##a*(b*c)=(a*b)*c## I need to have? 1) ##e*(e*e)=e*e=e##...
  21. E

    Discovering Associative Functions: A Prime Puzzle for Homework

    Homework Statement Let ##d(n)## denote the least prime factor of a positive integer ##n##, and let ##p## and ##q## be prime numbers. Find all functions ##f## such that ##d(f(p,q))## is associative for all ##p## and ##q##. Homework Equations ##f:\Bbb{P}\times \Bbb{P}\to \Bbb{P}## is a...
  22. G

    Prove an Associative Law (set theory)

    Homework Statement Prove that: A\cup(B\cupC) = (A\cupB)\cupCHomework Equations The Attempt at a Solution I never had to prove anything but I'll try. A\cup(B \cupC). Take: A = {1, 2, 3, 4, 5}, B = {5, 6, 7, 8, 9, 10}, C = {7, 8, 9, 10} (B\cupC) = P If A\cupP means A, united with the union of B...
  23. ArcanaNoir

    Show this function is associative (or provide a counter example)

    Homework Statement I'm trying to show that f(x,y)=x\sqrt{1+y^2}+y\sqrt{1+x^2} is associative, or provide a counter example. (All variables are real numbers.)Homework Equations The Attempt at a Solution I have it down to needing to show that...
  24. B

    Associative, Commutative, and Distributive Laws for Congruence Classes

    Homework Statement We are supposed to prove that the addition and multiplication of congruence classes for integers are associative, commutative, and distributive. Homework Equations None The Attempt at a Solution To me, it seems like this would just be trivial as all these...
  25. P

    Associativity of Group Operation

    Homework Statement Im looking at this example and trying to figure out how they showed it was associative. They start out with x*y=x+y+1 then they add in z to show it is associative. x*(y*z)=x*(y+z+1)=x+(y+z+1)+1=x+y+z+2 I don't know how they go from this x*(y+z+1)=x+(y+z+1)+1 and...
  26. S

    Generalized Associative Law for Groups

    Prove the Generalized Associative Law for Groups (i.e. a finite sum of elements can be bracketed in any way). The proof is outlined in D & F. I just want to know whether or not one part of my proof is correct. Show that for any group G under the operation °, and elements a1,...,an, any...
  27. M

    Example of Commutative but Not Associative Binary Operation

    hi , I met lot's of binary operation which is associative and commtative and I also met lot's of binary operation which is associative and not abelian but is there an example for a binary operation which is commtative and not associative ? I don't remmber that I've met one likes this...
  28. M

    How can we check that this operation is associative ?

    hello , there is an example here say that the binary operation which we denote it Δ is associative . Δ is defiened as follows : if A,B are two sets then AΔB=(A-B)U(B-A). and Δ is associative . but the proof is omitted . and I need to know the proof .
  29. S

    Halmos-Generalized Version of Associative Law

    I posted this in the homework section, but I think it probably belongs here. So Halmos says in Section 9 on families, "Suppose, for instance, that {Ij} is a family of sets with domain J, say; write K=UjIj and let {Ak} be a family of sets with domain K. Is it then not difficult to prove that...
  30. H

    Associative Property for Power Towers?

    A power tower (x^^n) is a variable raised to the power of itself n amount of times. x^^4 = x^x^x^x x^^3 = x^x^x x^^2 = x^x x^^1 = x I was wondering if an associative property for power towers exists. Does x^(x^x) equal the same thing as (x^x)^x? Is x^(x^^n) equal to x^^(n + 1)? If anybody...
  31. lpetrich

    C/C++ Associative Arrays in C/C++ - Favorite Strategies?

    Associative Arrays in C/C++ -- Favorite Strategies? I mean by that something like what Perl or Python or Ruby have, a.k.a. dictionary types. I can do it in Mathematica with the help of label[key] = value and Head and DownValues, but it's C/C++ where it is the most difficult of all. I once...
  32. D

    Associative Property of Convolution?

    Hi, I have a quick question about certain algebraic properties of convolution. If I have 3 functions x(f), y(f) and z(f), is the following true? [x(f) . g(f)] * z(f) = [x(f) * z(f)].g(f) I looked on Wikipedia but there's only a property like this if one of the terms is a scalar, so most...
  33. S

    Is this correct proving function addition is associative?

    Is this correct...proving function addition is associative? Homework Statement Let F be the set of all real-valued functions having as domain the set ℝ of all real number. Prove that function addition + on F is associative. Homework Equations The Attempt at a Solution I'm...
  34. K

    Linear Algebra - Associative property

    Homework Statement G=\{x\in R|0\leq x<1\} and for some x,y\in G define x*y=\{x+y\}=x+y-\lfloor x+y \rfloor Homework Equations The Attempt at a Solution I want to proof Associative property: x*(y*z)=(x*y)*z \Leftrightarrow x*(y+z-\lfloor y+z \rfloor)=(x+y-\lfloor x+y \rfloor)*z \Leftrightarrow...
  35. P

    Prove commutative and associative

    Homework Statement a function 'd' is a closed binary operation on a set called 'T'. There is an identity element named j. for all elements a, b, and c in the set 'T', we have d(a, d(b,c)) = d((a,c), b) can anyone help me show that d is commutative and associative? Homework...
  36. J

    Proving a set under an operation is associative. (Abstract Algebra)

    Homework Statement I'm trying to prove that this is a group. I already established elsewhere that it is a binary operation, so now I am onto proving associativity. I've tried many examples and so I'm confident it is associative, but now I just have to prove that.The Attempt at a Solution...
  37. S

    Associative property of convolution

    Hi There The associative property of convolution is proved in literature for infinite interval. I want to prove the associative property of convolution for finite interval. I have explained the problem in the attached pdf file. Any help is appreciated. Regards Aman
  38. P

    Find an Item Stored in Memory: Assoc. Memory & Time Reduction

    Time Required to find an item stored in memory can be reduced considerably if stored data can be identified for access by content of data rather than by an address please explain me how...?
  39. H

    Set Associative Cache: 8-Way Determination

    If I have the addressing format: Tag: 31-12 Index: 11-6 Byte offset: 5-0 And if my cache size is 32KB = 32*2^10 = 2^5*2^10 = 2^15 then this would represent a how many-way set associative cache? i.e., an x-way associative cache? Since bits 11-6 are used for the index, then this...
  40. F

    Associative Law for Multiplication - Probability Theory and Boolean Algebra

    Hi, I'm having some trouble solving one of the problems from my homework assignment. Homework Statement Prove: P((AB)C) = P(A(BC)) Where A,B,C are either true or false. Homework Equations We can't do this by using a truth table, we can use the following equations: P(A + B) = P(A) + P(B) -...
  41. N

    Cross product associative triples

    cross product "associative triples" Homework Statement We know that the cross product is not associative, i.e., the identity (1) (\vec{a}\times\vec{b})\times\vec{c} = \vec{a}\times(\vec{b}\times\vec{c}) is not true in general. However, certain special triples \vec{a};\vec{b};\vec{c} of...
  42. Saladsamurai

    Show that Matrix Multiplication is Associative

    Homework Statement Show that (AB)C=A(BC) I am just trying to do this to try to gain some experience with problems like this. I saw in my text that they did a similar example for distributivity using the definition of matrix multiplication, so I thought I could use that approach. The Attempt...
  43. P

    Question about proof of associative law for sets

    Hello, Trying to go through Tom Apostle text on Calculus. There is an exercise about proving the associative law for sets: So, (A U B) U C = A U (B U C) So, if we assume x to be an element in set in left hand side, than we can say x belongs at least to either A, B or C which in turn...
  44. K

    Establish associative law of multiplication

    Homework Statement Establish associative law of multiplication by considering absolute values and arguements. z1(z2z3)=(z1z2)z3 Homework Equations The Attempt at a Solution I think I need to use r(costheta +isintheta)...
  45. Saladsamurai

    Prove whether Cross-Product is associative

    Homework Statement Using the definition \bold{A}\times \bold{B}=AB\sin\theta show whether or not (\bold A\times\bold B)\times\bold C=\bold A\times(\bold B\times\bold C) I know this is probably easy, but I am missing the obvious here. So I started like this: Assuming \theta lies...
  46. L

    Associative Law help pleasethanks

    Associative Law...help please..thanks! b1. Homework Statement [/b] On the set of real numbers R, the following is defined *:RxR arrow R (x,y) arrow x*y=a(x+y)-xy find all the values of the real parameter a such that the operation is associative Homework Equations associative...
  47. P

    Prove that quaternions are associative

    Homework Statement prove that quaternions are associative. ie (qr)s = q(rs), where q,r,s are quaternions This isn't really a HW problem since I'm just wondering if there's a simpler way to prove associativity than the method I tried below Homework Equations i^2 = j^2 = k^2 = -1...
  48. O

    Convince a classmate that division is not associative.

    Homework Statement How would you convince a classmate that division is not associative? By "associative" the book aims at the Associative properties of multiplication and addition. Is this equation correct? a/(b/c) = (a/b)/c Homework Equations a + (b + c) = (a + b) + c a(bc) = a(bc)...
  49. A

    G is Finite and Closure Under Associative Product: Proving Group

    Given a finite set G is closed under an associative product and that both cancellation laws hold in G,Then G must be a group. I need to prove that G must be a group, I understand that for this I only need to show that : 1) There exist the identity 2) There exist the inverse. But...
  50. F

    How Can Associative Semantic Nets Revolutionize Business and Research?

    Is anybody here interested in Associative Semantic Nets? .