It's a homomorphism that's a bijection
What is an isomorphism?
When we assume that the statement we are proving is false and then everything gets all wacky and goofy
What is proof by contradiction?
A function f: A->B is this if for every element b in B there exists element a in A with f(a)=b
What is surjective?
One of them relations on a set that's reflexive, symmetric, and transitive.
What is an equivalence relation?
Proof that if G is an abelian group then for all a, b in G and for any positive integer n we have (ab)ⁿ=aⁿbⁿ
What is:
We proceed by induction on n.
Base case: n=1
(ab)¹=ab=a¹b¹
Inductive Hypothesis: We assume (ab)ⁿ=aⁿbⁿ
Inductive Step: We want to show that (ab)ⁿ+¹=aⁿ+¹bⁿ+¹
Because G is abelian and because we assumed (ab)ⁿ=aⁿbⁿ we have:
(ab)ⁿ+¹=ab(ab)ⁿ=a(ab)ⁿb=aaⁿbⁿb=aⁿ+¹bⁿ+¹
We conclude that (ab)ⁿ=aⁿbⁿ for any positive integer n.
What is a Diophantine equation?
Proof that n³-n+3 is divisible by 3 for all positive integers n using mathematical induction
What is:
Base Case: n=1
(1)³-1+3=3. 3 is divisible by 3.
Inductive Hypothesis: We assume that n³-n+3=3k, where k is some integer.
Inductive Step: We want to show that (n+1)³-(n+1)+3=3j where j is some integer.
(n+1)³-(n+1)+3=n³+3n²+2n+3=(n³-n+3)+3n²+3n
We assumed in our inductive hypothesis that n³-n+3=3k so we have
3k+3n²+3n=3(k+n²+n)
Therefore, (n+1)³-(n+1)+3 is divisible by 3 so we have proven that the n+1 case works. We conclude that n³-n+3 is divisible by 3 for all positive integers n.