Talk:Distributive property

Opening heading

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 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 numbers, and rounding does not have Distributive property. I was looking for an online reference that could explain this well in language that accountants (not mathamaticians) could understand. Could you help me with this? Is there an article on Non-distributivity? Or could you perhaps provide examples of things that do not have Distributive property? Thanks. -Sandy —The preceding unsigned comment was added by 71.138.248.226 (talk) 00:12, 2 January 2007 (UTC).

I've stuck a few examples into the article (section Distributivity and rounding). Tell me if that helps; if you can find some good references it'd be good to have them here. –EdC 02:32, 2 January 2007 (UTC)

This certainly deserves an explanation but the article is supposed to explain a piece of theoretical mathematics -- not a practical computational quirk. --90.185.76.189 (talk) 04:42, 29 January 2009 (UTC)

Division

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.

Thanks! Cherry Cotton 22:53, 25 April 2007 (UTC)

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. –EdC 20:40, 26 April 2007 (UTC)

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)

I don't think so, never learned it yet. I'd listen to the above user. Lunakeet 12:05, 2 May 2008 (UTC)

Image

The bottom part is the above multiplication carried out in two different parts. The top and bottom are equivalent.

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 taking place (4 * 2 = 8?) and where's the distributive law? - Simeon (talk) 18:29, 1 June 2008 (UTC)

Request move

I do not believe that "distributivity" is a word. I suggest moving this page to Distributive property. I don't ever recall seeing "distributivity" in any mathematics journal. Cliff (talk) 03:24, 28 April 2011 (UTC)

Move?

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: page moved. Vegaswikian (talk) 22:02, 27 June 2011 (UTC)

---

Distributivity → Distributive property –

• 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: Associativity has moved to Associative property, and Commutativity has moved to Commutative property. A thread was started on the current talk page to discuss the move, but no discussion has ensued. Cliff (talk) 03:04, 20 June 2011 (UTC)
• Google search for "distributivity" (excluding "distributive") finds about 244,000 results. That looks quite a lot like it being an accepted word. (And move Associative property and Commutative property back to "-ivity"?) Anthony Appleyard (talk) 04:55, 20 June 2011 (UTC)

• Support for consistency. –CWenger (^ • @) 05:11, 20 June 2011 (UTC)
• Support. A straightforward case. (Just for the record though, it is a word in use. Current OED'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.) Noetica^Tea? 23:11, 23 June 2011 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Distributivity of Conjunction over Exclusive Or

if and distributes over xor then

p and (q xor r) is (p xor q) and (p xor r)

However, this equivalence does not hold. Counter Example:

p := false

q := true

r := true

In this case,

p and (q xor r) is

false and (true xor true) 

is false and false

is false

but,

(p xor q) and (p xor r) is

(false xor true) and (false xor true) is

(true and true)

is true.

And does NOT distribute over exclusive or.

This result obviously also holds for the claimed distributivity of intersection over symmetric difference. Namely, Intersection does not distribute over symmetric difference.

--Xor logician (talk) 05:03, 14 September 2011 (UTC)

You distributed incorrectly to start. You should get...

(p and q) xor (p and r)

if and distributes over xor. Cliff (talk) 09:39, 17 September 2011 (UTC)

Distribution of disjunction over disjunction

$(P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\land (P\lor R))$

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. Ivashikhmin (talk) 10:49, 13 April 2012 (UTC) this is stupid bye now

Scalar multiplication example is missing

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.

Other examples of "mixed" distributive laws may abound. But this is a rather obvious one. Kluto (talk) 03:41, 10 September 2014 (UTC)

Antidistributivity

Currently the section Distributive property#Notions of antidistributivity says

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:[3]

(a ∨ b) ⇒ c ≡ (a ⇒ c) ∧ (b ⇒ c)

(a ∧ b) ⇒ c ≡ (a ⇒ c) ∨ (b ⇒ c)

These two tautologies are a direct consequence of the duality in De Morgan's laws.

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 right side implies the left side, but not vice versa, so they are not equivalent statements.

Comments, anyone? Loraof (talk) 00:13, 31 August 2017 (UTC)

We have to be careful not to mix formal logic and causality. See Material implication (rule of inference) and Material inference.

Let me try. In classical logic, the material conditional a ⇒ b is logically equivalent to ¬(a ∧ ¬b) which is logically equivalent to ¬a ∨ b. To find the matching Venn diagram for ¬a ∨ b, let us consider two sets A and B, and let us interprete a as x ∈ A, and b as x ∈ B. Using this, the logical expression ¬a ∨ b would be represented as the set A^C ∪ B, where A^C is the absolute complement of A.

Applying this to the second equivalence above, the left hand side (a ∧ b) ⇒ c would be represented as (A ∩ B)^C ∪ C, (with C a third set, and interpreting c as x ∈ C) and the right hand side would be represented as (A^C ∪ C) ∪ (B^C ∪ C), i.e., A^C ∪ B^C ∪ C. Now, De Morgan's laws say A^C ∪ B^C = (A ∩ B)^C, and so we are finished. Does this help?

Tea2min (talk) 08:22, 1 September 2017 (UTC)

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 neither (a ⇒ c) nor (b ⇒ c) is satisfied. Loraof (talk) 18:40, 1 September 2017 (UTC)

The statement a ⇒ c is false if and only if a is true and b is false, and true otherwise. Likewise, the statement x ∈ A ⇒ x ∈ C is false, for a given x, if and only if x ∈ A and x∉ C, and true otherwise. The statement x ∈ A ⇒ x ∈ C does not mean "for all x, x ∈ A ⇒ x ∈ C".

Consider the statements x ∈ A, x ∈ B, and x ∈ C. Let x = 1, to make x ∈ A true and x ∈ B and x ∈ C false. Now 1 ∈ A is true and x ∈ C is false, and then 1 ∈ A ⇒ 1 ∈ C is false. But 1 ∈ B is false, and then 1 ∈ B ⇒ 1 ∈ C is true (ex falso quodlibet), and then the combined statement (1 ∈ A ⇒ 1 ∈ C) ∨ (1 ∈ B ⇒ 1 ∈ C) is true. Similarly for x = 5.

– Tea2min (talk) 07:34, 2 September 2017 (UTC)

Got it! Thanks for your very clear explanations. Loraof (talk) 17:19, 2 September 2017 (UTC)

Vector spaces and generalizations to several sets

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.

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 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.

Im looking for a general definition to distributivity - a correction to these articles - that satisfies this dilemma. 50.35.97.238 (talk) 16:28, 24 November 2018 (UTC)