SOLUTION: How do you differentiate the properties associative,communitive, and distributive

Algebra ->  Test -> SOLUTION: How do you differentiate the properties associative,communitive, and distributive       Log On


   

Question 487933: How do you differentiate the properties associative,communitive, and distributive
Found 2 solutions by MathLover1, richard1234:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
The "Commutative Laws" say you can swap numbers over and still get the same answer ...
... when you add (a + b = b + a) or when you multiply (a * b = b * a)
Commutative+Laws:

a + b = b + a
a * b = b * a

The "Associative Laws" say that it doesn't matter how you group the numbers (i.e. which you calculate first) ...
... when you add (a + b) + c = a + (b + c) or when you multiply (a * b) * c = a * (b *c)
Associative+Laws:
(a + b) + c = a + (b + c)
(a * b) * c = a * (b * c)


The "Distributive Law" is the BEST one of all, but needs careful attention.
This is what it lets you do:
3*(2+4)
the 3* can be "distributed" across the 2+4, into 3*2 and 3*4


the "Distributive Law" says:
you get the same answer when you:
multiply a number by a group of numbers added together, or
do each multiply separately then add them


Distributive Law:
a * (b + c) = a * b + a * c
These laws are to do with adding or multiplying, not dividing or subtracting.
The Commutative Law does+not work for division:
Example:
12 / 3 = 4, but
3 / 12 = ¼
The Associative Law does+not work for subtraction:
Example:
(9 – 4) – 3 = 5 – 3 = 2, but
9 – (4 – 3) = 9 – 1 = 8
The Distributive Law does+not work for division:
Example:
24 / (4 + 8) = 24 / 12 = 2, but
24 / 4 + 24 / 8 = 6 + 3 = 9

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
For any binary operation * (not necessarily addition or multiplication):

* is said to be associative if (a*b)*c = a*(b*c) for all a,b,c ∈ S (S is a set composed of all possible a,b,c)

* is said to be commutative if a*b = b*a for all a,b ∈ S

* is said to be distributive over another operation (I'll denote this operation by &) if a*(b&c) = (a*b)&(a*c) for all a,b,c ∈ S.

For example, vector addition is commutative and associative, scalar products are distributive, the dot product of vectors is commutative, and the cross product of vectors is associative over vector addition, but not commutative (in fact, it is said to be anti-commutative, similar to subtraction). These are just examples of operations other than addition or multiplication that obey these properties.