Proof by contradiction means that you start by assuming the opposite of what you wish to prove, and then logically arriving at a contradiction. The existence of a contradiction implies that your premises were false, and since your premise was the opposite of what you wanted to prove, what you wished to prove must be true.
Proof by induction works by showing that a particular case (usually the first/smallest case) is true, and then showing that the fact that one case is true implies that the next case is also true. In other words, prove that your statement is true for n=1, and then prove that if the statement is true for m, it is also true for m+1. Since it's true for n=1, that implies it's true for n+1=2, and since it's true for n=2, that implies it's true for n+1=3, and so on, and thus you have proven that it's true for all n larger than your original case. Of course, it doesn't have to be that n implies n+1 -- it can be n-1, or any other variant, the point is that it chains up and leads to all values being proven.
Comments
Yes, completely different.
Proof by contradiction means that you start by assuming the opposite of what you wish to prove, and then logically arriving at a contradiction. The existence of a contradiction implies that your premises were false, and since your premise was the opposite of what you wanted to prove, what you wished to prove must be true.
Proof by induction works by showing that a particular case (usually the first/smallest case) is true, and then showing that the fact that one case is true implies that the next case is also true. In other words, prove that your statement is true for n=1, and then prove that if the statement is true for m, it is also true for m+1. Since it's true for n=1, that implies it's true for n+1=2, and since it's true for n=2, that implies it's true for n+1=3, and so on, and thus you have proven that it's true for all n larger than your original case. Of course, it doesn't have to be that n implies n+1 -- it can be n-1, or any other variant, the point is that it chains up and leads to all values being proven.