Here we review the method of proof by induction from CMPSCI 250.
Please answer the questions *during the discussion period*.

**Question 1:**The family of undirected graphs {K_n: n≥ 1} is defined as follows. K_{1}is a single node with no edges. K_{n+1}consists of a copy of K_{n}plus one new node, with an edge from the new node to each of the old nodes. Prove by induction on all n≥1 that there are exactly n(n-1)/2 edges in K_{n}.**Question 2:**A*forest*is an undirected graph with no cycles. For all n, prove by induction on all e from 0 through n-1 that a forest with n nodes and e edges has exactly n-e connected components.**Question 3:**A*full rooted binary tree*or*FBRT*of depth d, with d≥0, is defined as follows. An FRBT of depth 0 is a single node (its root) with no edges. An FBRT of depth d+1 consists of two FBRT's of depth d plus a new node, its root, and new directed edges from the root to the roots of the depth-d trees. Prove by induction on d that a depth-d FBRT has exactly 2^{d+1}-1 nodes and exactly 2^{d+1}-2 edges.**Question 4:**You are proving on HW#1 that if a and b are non-negative integers with a > b, and c is the remainder when a is divided by b, then gcd(a,b) = gcd(b,c). Using this fact, prove by strong induction on the number a+b that the Euclidean algorithm terminates with the correct greatest common divisor when started on inputs a and b. Assume the following recursive Java-like code is being used for gcd:`int gcd (int a, int b) { //assumes a > b >= 0 if (b == 0) return a; else return gcd (b, a%b);}`

Last modified 10 September 2003