01204211/activity4 counting 1

This is part of 01204211-58

In-class activities

A.1 Prove that the number of subsets of a set with $n$ elements is $2^{n}$ by induction.

A.2 (LPV1.3.2) Let $A=\{1,2,3,\ldots ,n\}$. What is the number of subsets of $A$ that contains $n$?

A.3 (LPV1.3.3) Prove that a nonempty set has the same number of odd subsets as even subsets.

For example, consider set $\{1,2,3\}$. It has 4 odd subsets: $\{1\},\{2\},\{3\},\{1,2,3\}$; and 4 even subsets: $\emptyset ,\{1,2\},\{1,3\},\{2,3\}$

Hint: show that a bijection exists.