Commutativity: Order Doesn't Matter

When you add two things, the order in which you add them doesn't matter (2+4 is the same as 4+2). The name of this property is the

*commutative*property. Examples of commutative operations are [note 1]:

- Multiplication: 2×3 is the same as 3×2
- Preparing a bowl of Cereal: it doesn't matter what order you add the milk and the cereal
- Putting on shoes: It doesn't matter what shoe you put on first
- Rotation: rotate left then right or right then left , you end up at the same place
- The logical operator
*or:*"Him or her" is the same as "Her or him." - Vector addition:
- The intersection of sets: :
- Multidimensional Integration:

Sometimes the order of things

Commutativity and associativity are important mathematical concepts. In part 2, we'll be looking at something associative and non-commutative. In part 3, We'll look at something commutative and non-associative.

[Notes][1] Some of my examples come from Wikipedia, some of them I heard before (in a classroom or in a conversation), some of them I just made up.

*do*matter. For example, division: is*not*the same as (the first is 2 the second is one half). When an operation does not commute, we call it non-commutative (that's simple enough). Some examples of non-commutativity are:- Subtraction: 5-3 is 2, but 3-5 is -2
- Division:
- Exponentiation:
- Matrix Multiplication: [see note 2]
- Making a cake: mix ingredients and
*then*bake, order matters - Putting your clothes on in the morning: underwear first, then pants
- Material Implication: is not the same as
- Rotations on a Rubik's Cube: turning the top then the right is different than turning the right then the top:

One important note on non-commutativity is that the operation doesn't have to be non-commutative on everything for it to be non-commutative. Take exponentiation for example, and are both 16. (Or if you're less mathematically inclined, use the examples of putting on your clothes: it doesn't matter if you put your pants on first, or your shirt.) An operation is still non-commutative if there's at least one pair of things that doesn't commute.

Associativity: I Don't Care Who You Hang Out With

When adding three (or more) numbers, it doesn't matter how you group those numbers. For example, if you wanted to add 2, 3 and 5, you could add 2 and 3, then add 5, or you could add 2 to 3 plus 5. This is much clearer with notation: 2+(3+5) is the same as (2+3)+5. The way you group the numbers doesn't change the answer. This is called this associativity. Other examples of associativity include:

- Multiplication: 2×(3×4) is the same as (2×3)×4
- Mixing trail mix: mixing raisins and granola then adding walnuts is the same as mixing raisins into a granola/walnut mixture
- Translations on a plane: going up 4, then going to the left 3 and down 2 is the same as going up 4 and left 3 then going down 2
- Maximum function: max(a, max(b,c)) = max(max(a,b),c)
- The logical operator
*and:* - Composite functions:
- Putting together a puzzle:

You probably guessed it; some things are non-associative. The best example is division: is 1, but is 4. When dividing, the way you group things matters. Other non-associative examples include:

- Subtraction : (8-5)-2 = 1 but 8-(5-2) = 5
- Exponentiation:
- Doing the dishes: Washing and drying dishes, then putting them away is definitely not the same as washing dishes that are dried and put away
- English Expressions: A
*(smart dog) owner*is someone that owns a smart dog, a*smart (dog owner)*is someone that is smart and owns a dog. - Material implication: is not the same as
- The one dimensional distance formula
*d*(*x*,*y*) = |*x*-*y*|:

Commutativity and associativity are important mathematical concepts. In part 2, we'll be looking at something associative and non-commutative. In part 3, We'll look at something commutative and non-associative.

[Notes]

[2]

## No comments:

## Post a Comment