How Do You Prove (ab)^-1 Equals a^-1b^-1?

[jgens]

Homework Statement

Prove (ab)^-1 = a^-1b^-1 for all a,b =! 0.

Homework Equations

Multiplicative inverse property: (a)(a^-1) = 1
Commutivity: ab = ba
Associativity: (ab)c = a(bc)
Transitivity: If a = b and b = c then a = c

The Attempt at a Solution

Early today, my friend and I had a discussion regarding this statement and its proof using the principles stated above. The proof I provided is as follows:

(ab)(ab)^-1 = 1 and 1 = (a)(a^-1)(b)(b^-1); therefore it follows by transitivity that (ab)(ab)^-1 = (a)(a^-1)(b)(b^-1). By commutivity and associativity it follows that (ab)(ab)^-1 = (ab)(a^-1)(b^-1), ultimately yielding (ab)^-1 = (a^-1)(b^-1).

The proof my friend posted is as follows:

(ab)^-1 = 1/(ab) = (1/a)(1/b) = (a^-1)(b^-1)

I argued that my friend's proof was not valid as it relies on the unproven (yet true) assumption that (ab)-1 = 1/(ab) which he would need to demonstrate was true before using it in a proof. I think my friend is right on this one and my criticisms aren't truly valid; however, I thought I would check here with people who are familiar with mathematics to ensure that any of the proofs, criticisms, etc. are valid or invalid.

Thanks!

 

[quantumdude]

Your proof is correct, and your friend's isn't. These proofs should only refer to addition, multiplication, and the axioms. Division is defined later in terms of multiplication. So things like $1/(ab)$ shouldn't even be showing up here.

 

[Architeuthis Dux]

Both proofs are valid and correct. Your friend's proof is simply a more direct and concise way of proving the statement. It is true that the assumption that (ab)-1 = 1/(ab) is not explicitly proven, but it is a well-known property of multiplicative inverses that can be easily verified using the multiplicative inverse property. In fact, your proof also relies on this property, as you use the fact that (ab)-1 = (a)(b)-1 in your transitivity step. Therefore, both proofs are valid and your criticisms are not entirely valid. It is always important to check and verify assumptions used in proofs, but in this case, the assumption is a well-known property that can be safely used.