00:01
So this question asks to prove that all sets with n elements have 2 to the n subsets, where the set itself and the empty set are considered subsets.
00:09
So we need to do this by proof of induction, prove by induction.
00:14
So we need to prove the base case, which is to prove that the statement is true for the lowest possible number, for the lowest possible value of n, so that's n equal to 1.
00:26
And then we need to prove the inductive case, which is to prove.
00:31
That for any integer k which satisfies the statement or which makes the statement true k plus 1 also makes the statement true so let's first prove the base case let's construct a set called that set 1 or s1 let's construct a set s1 which has one element let's say it has the element 1 this can be broken down into two elements the set itself and the empty set.
01:19
So we know that s1 only has two possible subsets, and the number of subsets should equal 2 to the n, so that's 2 to the 1, and that's equal to 2.
01:43
So we've proven the base case is true, so when n equals 1, the statement is true.
01:51
So for the inductive step, we can break this down into two steps.
01:56
So forming the inductive hypothesis is first.
01:59
So this is to assume that for any n equals k, so where k is any integer, the statement also holds.
02:12
So we can track the set sk, and that can have two to the k subsets...