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we define two binary operations called field multiplication and field addition. We also define a fourth function, an automata to be precise, called scalar multiplication which operates between the field elements and the vector elements producing vectors. We require "left distributivity" of scalar multiplication over vector addition. We also require "right distributivity" of scalar multiplication over field addition. These arent real distributivity, as per this articles definition, because it operates on elements of two different sets, rather than being two operations on a single set. Additionally, in the case of "right distributivity", the field addition turns into vector addition in the process; some sort of isomorphic mapping. This is one more problem with the distribution definition.
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In the definition of the vector space (and possibly inner product), two sets are utilized. In the case of vector spaces, one set is the set of vectors and the other is the set of scalars (which is a field). In the vectors we define a single binary operation called vector addition, and in the scalars
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Despite the citation to a book (which I don't have access to), the second of these seems to be false. In a Venn diagram, the left side says that the intersection of A and B is entirely within C. But the right side says that at least one of A or B is entirely within C, which is not the same thing. The
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Thanks. Maybe I misunderstand what is meant by (a ⇒ c) ∨ (b ⇒ c). Doesn't it mean "either a implies c or b implies c or both"? Suppose A = {1, 2, 3}, B = {3, 4, 5}, C = {2, 3, 4}. Then the left side (a ∧ b) ⇒ c is satisfied, since 3 is in C. But 1 is in A but not in C, and 5 is in B but not in C; so
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I've removed this image because I think it's not very clear.. I've been looking at it but I've yet to see two multiplications taking place there.. there are four and eight Xs above the line and some more below it but I fail to see how this image clarifies the distributive law. What multiplication is
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I'd like to see this article be more clear on whether or not division is distributive over addition. It is, isn't it? But, I was confused about it from reading this article. Distributivity of division over addition is mentioned and invoked as though it were the case, but isn't explained to be so.
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Division (in the reals) is simply multiplication by the reciprocal (the multiplicative inverse) of the divisor, so distributivity follows immediately. The same holds for the distributivity of multiplication (or division) over subtraction.
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The article on vector spaces clearly describes two operations as "distriubtions" but the article on distribution uses a narrower definition that doesnt apply. There is a contradiction between articles.
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In the study of propositional logic and
Boolean algebra, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:
708:'s earliest citation is from 1940: "W. V. Quine Math. Logic 60 Distributivity always justifies an operation analogous to that of 'multiplying through' in arithmetic." Quine? That's pretty canonic.)
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sum of the original numbers. They have to start moving pennys around and they blame their electronic spreadsheets. We all know that division has
Distributivity, but the spreadsheet also rounds the
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Distributivity of scalar multiplication wrt addition of scalars resp. vectors is mentioned explicitly on the Vector Space page. Clearly it is an important example, so it should be mentioned.
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This result obviously also holds for the claimed distributivity of intersection over symmetric difference. Namely, Intersection does not distribute over symmetric difference.
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The latter is a redirect to the current article. The current title is not a word, nor is it commonly used in mathematical journals. This move will match recent moves:
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You have to be careful though, because it only works the one way: (1 + 5)/7 = 1/7 + 5/7, but 1/(5 + 7) ≠ 1/5 + 1/7. Hope this helps. 10:11, 24 June 2007 (UTC)
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I have many associates in the accounting field who are baffled when they divide a list of numbers and the sum of those results does not match the dividing the
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Google search for "distributivity" (excluding "distributive") finds about 244,000 results. That looks quite a lot like it being an accepted word. (And move
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This certainly deserves an explanation but the article is supposed to explain a piece of theoretical mathematics -- not a practical computational quirk. --
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This formula is wrong. E.g. if P=false, Q=true, R =false then left side of the formula is true while right side is false.
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Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.
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Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.
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Im looking for a general definition to distributivity - a correction to these articles - that satisfies this dilemma.
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The bottom part is the above multiplication carried out in two different parts. The top and bottom are equivalent.
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right side implies the left side, but not vice versa, so they are not equivalent statements.
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Other examples of "mixed" distributive laws may abound. But this is a rather obvious one.
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These two tautologies are a direct consequence of the duality in De Morgan's laws.
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I do not believe that "distributivity" is a word. I suggest moving this page to
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I don't think so, never learned it yet. I'd listen to the above user.
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Applying this to the second equivalence above, the left hand side (
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You distributed incorrectly to start. You should get...
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