Here are silly but correct intuitive definitions for properties of binary operations. These properties are essential for understanding complex elements of abstract algebra such as groups, rings and fields thus it is important to have strong intuitive binding for fundamental definitions.
Let’s make a breakfast with coffee and sandwich!
Please be careful when choosing the sort of coffee and milk. For successful understanding you must be capable of taking at least four cups of coffee. Also choose carefully a sort of cheese since you will have to roast it in an oven for 5 minutes. I recommend you to use foil when roasting since cheese and tomato can be really sticky.
Commutativity
Formal definition for operation “+”:
a + b = b + a
Intuitive definition:
Exercise 1: take an empty cup add coffee then milk. Exercise 2: Take an empty cup add milk then coffee. Both cups now contain coffee with milk. Taste both cups. If you do everything right, the coffee would be tasty in both cups.
In groups theory commutativity is used to define abelian group. An abelian (or commutative) group is a group which operation is commutative. Since we don’t have a group (see below) we also don’t have abelian group.
Please note that we are not putting sugar at this time. So for those of you who cannot drink coffee without sugar I suggest to skip this exercise and go straight to the associativity recipe.
Associativity
Formal definition for operation “+”:
(a + b) + c = a + (b + c)
Intuitive definition:
You have a cup of coffee, milk and sugar. Exercise 1: Mix coffee with milk (a+b) and then add sugar (c). Exercise 2: Mix milk with sugar (b+c) and then put it all together with coffee (c). Practice by taking two cups of coffee with milk and sugar. Of course the taste should be great in both cups.
Associativity is required in group theory to define a “group”. Unfortunately we cannot define a group with elements like coffee, sugar and milk and operation of mixing because a group must have following properties.
- for all elements in a set, the result must also be in this set. This is fine since milk, coffee, sugar and the result are all in the same set of food ingredients.
- associativity is defined with operation of mixing.
- Identity element: here we have a problem since I cannot think of an element which will allow me to mix milk or sugar with it and still have milk or sugar.
- Inverse element: so far I cannot figure out such element which will allow me to mix it with milk and produce identity element defined above.
We have troubles defining identity element and inverse element so we cannot have a group.
If you are still hungry proceed to the distributivity sandwich.
Distributivity
Distributivity is little bit more complicated since it is defined for two binary operations.
Formal definition:
a * (b + c) = (a * b) + (a * c)
Intuitive definition:
Let's make some good sandwich! Our recipe requires one piece of bread, a tomato and a slice of cheese. One way to cook a perfect sandwich is to put tomato and cheese on bread and roast it all in oven for 5 minutes. Another way is to roast it separately: first bread, then tomato, then cheese. Before eating you still need to put roasted tomato and cheese on bread.
If you do this exercise right in terms of taste both methods are equal.
So we have:
"*roasting*" ("bread" + "tomato" + "cheese") = "*roasting*" "bread" + "*roasting*" "tomato" + "*roasting*" "cheese"
Distributivity is used to define rings and fields.
Good appetite!