Computer Science 261 Exercise set #6 Due: Thursday, Oct. 11

(Incorporating corrections made in class) Set Theory

Suppose that

• A = {a, b, d, e}

• B = {b, c, d, f}

• C = {r, s } calculate

• A ∩ B

• A ∪ B

• A − B

• C × A Truth Tables Write truth tables for the following:

• p → q

• ¬p → ¬q

• ¬q → ¬p

• ¬p ∨ q

• ¬(p ∧ q)

• ¬p ∨ ¬q

• ¬(p ∨ q) Digital circuitry uses nand (not and) and nor (not or) and inverter gates. We usually write 0 for false (or no current) and 1 for true (or 5v current). With this in mind, what will be the output of the circuit given by

• (p nand q) nor (not p) ?

• can you draw the circuit?

• can you construct a circuit that will give us ¬p ∨ q?

Induction

A.

• State the principle of mathematical induction

• It is a property of positive integers that any subset of positive integers has a smallest element in the subset (this is not true, for example, of rational numbers or real numbers). Use this fact to give an informal argument for the principle of induction. Hint: For a given statement about positive integers, consider the subset consisting of all positive integers for which the statement is false.

B. Prove the following statements using mathematical induction

• ∑_{k=1}^{n} k = \frac{n(n+1)}{2}

• ∑_{k=1}^{n} k^{2} = \frac{n(2n+1)(n+1)}{6}

• ∑_{k=0}^{n} 2^{k} = 2^{(n+1)} − 1

• ∑_{k=0}^{n} \frac{1}{2^{k}} = 2 − \frac{1}{2^{n}}

• ∑_{k=1}^{n} k2^{k} = (n − 1)2^{(n+1)} + 2

• Recall that Fibonacci numbers are defined with F_{0} = F_{1} = 1, F_{2} = 2, … and so on. if ϕ = ^{\frac{(1+\sqrt{5})}{2}}, prove that F_{n} ≥ ϕ^{n−1} for n ≥ 1 You can use the fact that ϕ^{2} = 1 + ϕ. An interesting exercise would also be to show that F_{n} ≤ ϕ^{n} and use this to make a guess about lim_{n→∞} \frac{F_{n+1}}{F_{n}}. (This question was done in class, and so will not be marked)