Thread: [SOLVED] Binary Operations

Suppose that two binary operations, denoted by and , are defined on a nonempty set S, and that the following conditions are satisfied .

(1) and are in S
(2) and
(3)

Also, and , the elements and are defined recursively as follows:


if and have been defined, then and

Which of the following must be true?

(i) and

(ii) and

(ii) and
This is one obviously true.

I am struggling with proving or disproving 1 and 2

Originally Posted by dwsmith

Suppose that two binary operations, denoted by and , are defined on a nonempty set S, and that the following conditions are satisfied .

(1) and are in S
(2) and
(3)

Also, and , the elements and are defined recursively as follows:


if and have been defined, then and

Which of the following must be true?

(i) and

(ii) and

(ii) and
This is one obviously true.

I am struggling with proving or disproving 1 and 2

I think (i) is false, but I don't think my demonstration constitutes a rigorous proof.

We can rewrite



Since is not commutative, we cannot rearrange terms.

I think (ii) is true and can be proven from the definition of multiplication and conditions (2) and (3).

We can rewrite



Now with commutativity and associativity we can simply rearrange terms to get

Why does mean ? I don't see how you deciphered that from what was giving.

Originally Posted by dwsmith

Why does mean ? I don't see how you deciphered that from what was giving.

Well





...

Originally Posted by dwsmith

This relies on



which I suppose is obvious since it's just the version of (iii), nevertheless it still requires proof.

The way I wrote it down, weak induction could be used for a formal proof.

I'm curious as to your obvious proof of (iii), since I would have used



from which (iii) follows trivially.

This is why 3 is obvious.