Question

. Let ∈ ℕ and A be a subset in some universal set U. Use

mathematical Induction to prove the following: If || = , then
|()| = 2ⁿ .

Answer 1

To prove |A^n| = 2^n using mathematical induction, we start by proving the base case and then assume the statement holds for k and prove it for k+1. We show that |A^k+1| = 2^(k+1) by using the fact that |A^k ∪ A| = 2^k + 2 = 2^(k+1).

Step-by-step explanation:

To prove that |A^n| = 2^n using mathematical induction, we will start by proving the base case. When n = 1, A^n represents the set A itself, so |A^1| = |A| = 2^1 = 2. Now let's assume that |A^n| = 2^n holds true for some natural number k, and prove that it also holds true for k+1.

Using the assumption, we have |A^k+1| = |A^k ∪ A|.

Since A^k is a set with 2^k elements, and A is a set with 2 elements, the union of these sets will have 2^k + 2 = 2^(k+1) elements. Therefore, |A^k+1| = 2^(k+1), which completes the proof.