Chapter Nine, Tutorial Four
Very Elementary Abstract Algebra

We finish by showing how the logic we've developed is the basis for mathematical reasoning. We take one example: Group Theory.

Group theory is inspired by numbers but is not about them in particular. Instead, it pertains to many things including numbers. The idea is that certain interesting features of numbers are also true of quite different types of object. (We'll see examples later in this tutorial.) Any of these different types, is called a "group".

Now, what we'll do first is "abstract" these interesting features (just using our possibly vague memory of 7th grade number theory!) and express them in PLIF sentences. Together these sentences will give the definition of a group. Finally, when the sentences are taken as premises, we can derive conclusions about any group.

Group theory can be stated with three "axioms" all of which involve a single two-place functor: '+'. This should make you think about addition: that will help you understand the axioms that follow. But we'll drop this way of thinking about '+' in a moment.) Also, we'll temporarily use the English/mathematical symbol for identity (it's much easier to read until we introduce some definitions). Then, the three premises (along with comments) are:

  1. (^x)(^y)(^z) x+(y+z) = (x+y)+z (The order of applying the '+' operation is immaterial)
  2. (%x)(^y) (x+y) = (y+x) = y (We may call the object x an identity element becuase it doesn't change y when '+' is applied to the two. We will prove that there is only one identity element for any group. Let's call it e.)
  3. (^x)(%y) (x+y) = (y+x) = e (y is the inverse of x: we will prove that there is only one such inverse for any x and call it -x.)

This is meant to be rather abstract. But to get a handle on what they mean, it's a good idea to think of a particular group: the set of integers {...-3,-2,-1,0,1,2,3...} along with the operation of addition.

That is, if we take the following interpretation...

Interpretation 1:

universe of discourse: the integers, {...-3,-2,-1,0,1,2,3...}
+: the addition function

...then we have an interpretation that makes each of the three statements true.

  1. (^x)(^y)(^z) x+(y+z) = (x+y)+z: We know that the order of addition doesn't matter in aritmatic, so 1 is true on the given interpretation.
  2. (%x)(^y) (x+y) = (y+x) = y: What is the number (in place of x) which doesn't change the value of a number y when added to that number? It's 0, of course. So, 0 is the "identity element" of the group that is interpretation 1.
  3. (^x)(%y) x + y = 0: For every integer x there is an opposite, it's negation, -x making this true.

However, this is just an example of a group! We've said that there are other groups as well. And some collections which are not groups.

For example, the integers with multiplication do not form a group! That is, reinterpret the '+' functor as multiplication:

Interpretation 2:

universe of discourse: the integers, {...-3,-2,-1,0,1,2,3...}
+: the multiplication function

It's OK to define our '+' symbol any way you'd like (as long as it's a two place function). So, for the moment try to think of '+' as multiplication! ('3+3=9' on the second interpretation.)

But, this second interpretaion is not a group. That is, it doesn't make all three of our premises true. Why not?

Click on the correct explanation of why the second interpretation is NOT a group:

  1. The first axiom does not apply: multiplication depends on order. (E.g., [ 3 times ( 4 times 2) ] is not equal to [ (3 times 4) times 2 ]
  2. The second axiom does not apply: there is no identity element for multiplication amoung the integers.
  3. The third axiom does not apply: there is no inverse element for multiplication amoung the integers.