11) Let u1; u2; u3; .... denote the Fibonacci numbers. Prove,by induction on n, that u5n is divisible by 5 for all positive integers n 

 

 

Math 109. Instructor: Chow
Homework #1. Due 3:00 pm on Tuesday, July 5, 2016.
Problem 1: Construct truth tables for the statements:
(i) not (P or (not Q))
(ii) (not P ) and Q.
(iii) Are the statements ‘not (P or (not Q)’and ‘(not P ) and Q’logically
equivalent? Explain.
Problem 2: Do Exercise 2.5(ii) on p. 20.
Problem 3: Prove that:
(i) n2 5n 6 = 0 () (n = 6 or n = 1):
(ii) n2 5n 6 > 0 () (n > 6 or n < 1): Problem 4: Prove that ‘(P and Q) ) R’and ‘(P ) R) or (Q ) R)’
are logically equivalent.
Problem 5: (i) De…ne an integer a to be happy if there exists an
integer b such that a = 3b + 1: Prove that the square of a happy integer is
happy.
(ii) De…ne an integer a to be silly if there exists an integer b such that
a = 3b + 2: Prove that the square of a silly integer is happy.
Problem 6: Using Axioms 3.1.2, do Exercise 3.8 on p. 29.
Problem 7: Using Problem 5, prove by contradiction that for any integer n,
n2 is a multiple of 3 ) n is a multiple of 3.
Problem 8: Are the statements ‘P ) (Q and R)’ and ‘(P or (not
Q)) ) R’equivalent?
Problem 9: Prove the following. Let a and b be real numbers.
If ab = 0, then a = 0 or b = 0:
Problem 10: Prove by induction on n that, for all positive integers n,
12 + 4 2 + + (3n 2)2 = n 6n2 3n
2 1 : Problem 11: Let u1 ; u2 ; u3 ; : : : denote the Fibonacci numbers. Prove,
by induction on n, that u5n is divisible by 5 for all n 2 Z+ . 1 Problem 12: (i) Prove by induction on n that n! > 4n for all integers
n such that n 9.
(ii) Is the following statement true? If n is an integer with n
3 and
n
n+1
n! > 4 , then (n + 1)! > 4
.
(iii) Why are your answers for parts (i) and (ii) consistent?
Problem 13: Let A and B be sets. Prove:
(i) A A [ B. (ii) A [ B = A if and only if B (iii) A \ Ac A. = ?. Problem 14: (De Morgan laws) Prove, using truth tables, that
(A [ B)c = Ac \ B c :
That is, construct a truth table whose columns are headed by:
x 2 A, x 2 B, x 2 A [ B, x 2 (A [ B)c , x 2 Ac , x 2 B c , x 2 (Ac \ B c ).
Which two columns need to have the same truth values? 2

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