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Sep
28
revised Use induction to prove that a function is not one to one
Further explained the process of restricting f and renamed variables.
Sep
28
comment Use induction to prove that a function is not one to one
By the way, the notation is defined this way: for all integers $N$, $[N]:=\{1,...,N\}$.
Sep
28
comment Use induction to prove that a function is not one to one
For induction, you prove the base case (here, when $k=1$). Then assume the theorem holds for all $k<K$ and show that implies that the theorem holds for the next value of $k=K+1$. The inductive hypothesis in the above proof is that there is no one-to-one function $f:[K+1]\rightarrow [K]$. We want to show that this lack of existence of a 1-1 function from [K+1] to [K] implies the lack of existence of a 1-1 function from [K+2] to [K+1]. Use a proof by contradiction. Suppose such a 1-1 function $f$ exists, and use it to construct a function that is 1-1 from [K+1] to [K], a contradiction.
Sep
28
comment Use induction to prove that a function is not one to one
Are you asking about the second to last sentence?
Sep
28
answered Use induction to prove that a function is not one to one
Aug
28
awarded  Tumbleweed
Aug
21
revised An example of $k$-independent distributions.
Added tags
Aug
21
asked An example of $k$-independent distributions.
Jul
20
awarded  Yearling
Jul
5
comment Sequence of natural numbers
I agree. The approach suggested by Tomas seems very promising.
Jul
5
revised Sequence of natural numbers
added 10 characters in body
Jul
5
answered Sequence of natural numbers
Jul
5
awarded  Critic
Jul
3
answered $A$ is a subset of $B$ if and only if $P(A) \subset P(B)$
Jan
26
awarded  Self-Learner
Aug
24
comment Theorems with an extraordinary exception or a small number of sporadic exceptions
Wouldn't theorems with no exceptions be in this list too? (Such as all integers are divisible by 1)?
Aug
19
answered What is the result of $5-0\times3+9/3=$?
Aug
19
accepted A modification of the harmonic series that causes it to converge.
Aug
18
comment A modification of the harmonic series that causes it to converge.
@J.M. great find! This is cool because I was working on proving that if one excludes any sequence of length n from appearing in the number, then the series still converges.
Aug
18
comment A modification of the harmonic series that causes it to converge.
Very nice and succint. You'll see that our methods are essentially the same. I wonder if there is another "good" way to solve this.