zakinutuzi

2021-12-15

Let

Let

Determine

Neunassauk8

Beginner2021-12-16Added 30 answers

Step 1

Bergman asks to derive a number of identities from the defining identities for$W$ , and these identities help to show that the universe of $bf{\left\{F\right\}}_{W}(x,\text{}y)$ is

$\{x,\text{}y,\text{}(x\cdot y),\text{}(y\cdot x)\}$ .

More precisely, Bergmans

Bergman asks to derive a number of identities from the defining identities for

More precisely, Bergmans

Shannon Hodgkinson

Beginner2021-12-17Added 34 answers

Step 1

To save typing, I'll write

Bergman gives in part (a) a list of identities which follow from the defining identities of

1)

2)

3)

4)

5)

For completeness, I'll give proofs of these five identities inside the spoiler blocks.

1)

2)

3)

4)

5)

Ok, now we add the additional defining identity of

Our task is to understand

Recall that every element of the free algebra

Level 0: a, b

Level 1: a, b, aa, ab, ba, and bb.

Note that we can eliminate aa and bb since they are redundant:

Now it turns out that any term at Level 2 (any product of a, b, ab, and ba) can be shown to be equivalent to a term at Level 1, using the identities above. They multiply according to the following Cayley table:

I've hidden the derivations in the spoiler blocks below.

0

22+64

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