1. Prove by (weak) mathematical induction that (+i | 1≤i≤n : 2i + 3) = n2 + 4n for all n ≥ 0.
2. Prove by (weak) mathematical induction that (+i | 1≤i≤n : 1/2i) = 1 - 1/2n for all n ≥ 0.
3. Prove by (weak) mathematical induction that 4n - 1 is divisible by 3, for all n ≥ 0.
Hint: If you divide 4n+1 - 1 by 4n - 1, you get a quotient of 4 and a remainder of 3. That is, 4n+1 - 1 = 4·(4n - 1) + 3
4. Define f : ℕ → ℕ as follows:
f.0 | = | 1 |
f.1 | = | 3 |
f.n | = | 4·f(n−1) − 3·f(n−2) (for n≥2) |
Prove by (strong) mathematical induction that f.n = 3n for all n ≥ 0.
5. Assume as true the claim that every acyclic graph that contains at least one edge also includes a vertex of degree one. Let G be an connected acyclic graph (i.e., a (free) tree). Prove by mathematical induction that v(G) = e(G) + 1 (i.e., the number of vertices in G is one more than the number of edges in G).
Procedure: Show that the statement is true in the case that v(G) = 1. That is the base case, or basis. Then take as an induction hypothesis (IH) that the statement is true for all trees having n vertices, where n≥1 is arbitrary. Let G be a tree having n+1 vertices. Use the IH in arguing that the statement is true for G. (That is the induction step.)
6.
* / \ / \ / \ * * / \ | / \ | * * * / \ / \ * * | / \ | / \ * * * | | | | * * |
Root is a leaf T0 |
Root has one child T1 |
Root has two children T2 |
---|---|---|
* |
* | | | | * / \ / \ / TC \ +-------+ |
* / \ / \ / \ / \ * * / \ / \ / \ / \ / TL \ / TR \ +-------+ +-------+ |
The height of a rooted tree is equal to the length of a longest path among all paths from the root to the leaves. Recursively, the height of a rooted tree is zero if the root is its only node and, otherwise, the height is one plus the height of its "tallest" proper subtree (which would be a subtree rooted at one of the children of the root).
Referring to the diagram above, we have
Prove by (weak) mathematical induction that, for all n≥0, every rooted binary tree having no less than 2n nodes has height no less than n.
To clarify, what is to be proved is that, for all n≥0 and all rooted binary trees T,
where v(T) is the number of nodes/vertices in T and h(T) is the height of T.
Hint: Regarding the induction step, suppose that T is a rooted binary tree having at least 2n+1 nodes. If T is of the form of T1, how many nodes must be in TC? If T is of the form of T2, how many nodes must be in larger among TL and TR?