4 ms·
2 books I recommend: how to prove it, and the book of proof, get a real analysis book, and if possible take a course. it took me around 5 years to be able to pr
by readme 4y ago
2 books I recommend: how to prove it, and the book of proof, get a real analysis book, and if possible take a course. it took me around 5 years to be able to prove stuff... you need relaxation, fluid thinking, and a breadth of knowledge of facts you can use to prove stuff
we can try your example
we want to show that (a^b)^c = a^bc
let's work on the LHS
by the definition of an exponent, we know that a^b is just a * a * ... * a b times, so we can rewrite it as:
(a_0 * a_1 * ... * a_b)^c
by the same definition, we can multiply the quantity inside the parenthesis by itself c times:
(a_0 * a_1...a_b)_0 * (a_0 * a_1 * ... * a_b)_1 * ... * (a_0 * a_1 * ... * a_b)_c
now, use the fact that a^m * a^n = a^(m+n) to consolidate the parenthesis, since each factor has an exponent of 1 we can use simple counting:
(a^b)_0 * (a^b)_1 * ... * (a^b)_c
Repeat the previous step c times, we end up with
a^(b_0 + b_1 + ... + b_c)
which of course is just
a^(bc)
therefore (a^b)^c = a^(bc)
- vessenes 4y agoside note; there are some subtleties here for the rationals and serious difficulties with this method for irrationals. I agree this is a good start for someone thinking about how to prove some simple math ’rules’ though, so please don’t take this as discouragement; rather encouragement to keep opening up minds when discussing math!
- giomasce 4y agoSuppose you have three finite sets A, B and C, each with a, b and c elements. Let A->B be the set of functions from A to B. Then it has b^a elements. Let AxB the set of couples with the first element in A and the second on B. Then it has ab elements. So to prove that (a^b)^c = a^(cb) you have to prove that there is a bijection between C->(B->A) and (CxB)->A. Ever heard of currying and uncurrying? (BTW, I didn't really use the hypothesis that the sets are finite, the proof is also valid for transfinites if you care about those)