Mathematical induction works if you meet three conditions:įor the questioned property, is the set of elements infinite?Ĭan you prove the property to be true for the first element? All the steps follow the rules of logic and induction. That step is absolutely fine if we can later prove it is true, which we do by proving the adjacent case of P(k + 1). Many students notice the step that makes an assumption, in which P(k) is held as true. Those simple steps in the puppy proof may seem like giant leaps, but they are not. You have proven, mathematically, that everyone in the world loves puppies. If you can do that, you have used mathematical induction to prove that the property P is true for any element, and therefore every element, in the infinite set.
Another way to state this is the property ( P ) for the first ( n ) and ( k ) cases is true:
So what was true for ( n )=1 is now also true for ( n )= k. Instead of your neighbors on either side, you will go to someone down the block, randomly, and see if they, too, love puppies. Your next job is to prove, mathematically, that the tested property P is true for any element in the set - we'll call that random element k - no matter where it appears in the set of elements. In the silly case of the universally loved puppies, you are the first element you are the base case, n. Proving some property true of the first element in an infinite set is making the base case.
Yet all those elements in an infinite set start with one element, the first element. In logic and mathematics, a group of elements is a set, and the number of elements in a set can be either finite or infinite. Before we can claim that the entire world loves puppies, we have to first claim it to be true for the first case.
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