HW4 Solutions

# HW4 Solutions - CSE 20 Discrete Mathematics Spring 2010...

This preview shows pages 1–2. Sign up to view the full content.

CSE 20: Discrete Mathematics Spring 2010 Problem Set 4 Solutions Instructor: Daniele Micciancio Due on: Thu. May 6, 2010 In this problem set all variables range over the set of nonnegative integers. Problems 2, 3 and 4 are on the well ordering principle which states that “Any non empty set of positive integers has a smallest element”, and was covered in class on Thursday April 28. Alternatively, you can solve the problems using induction, which you can ﬁnd in the textbook in unit IS, and will be covered in class on May 4. Problem 1 (6 points) Use the deﬁnition “a divides b” (namely, c.b = a · c ) to prove each of the following statements: 1. If x | a and x | b , then x | ( a + b ) Proof. x | a and x | b ⇒ ∃ c,d.a = x · c,b = x · d. ( a + b ) = x · c + x · d = x · ( c + d ) . Therefore x | ( a + b ) . 2. If x | a and x | ( a + b ) , then x | b Proof. x | a and x | ( a + b ) ⇒ ∃ c,d.a = x · c,a + b = x · d . b = ( a + b ) - a = x · d - x · c = x · ( d - c ) . Therefore x | b . Problem 2 (10 points) Remember the deﬁnition of even and odd: Even ( n ) ( k.n = 2 k ) and Odd ( n ) ( k.n = 2 k + 1) . For each of the following statements, formulate the statement using predicate logic and the predicates Even and Odd, and give a proof that the stamement is true: For any even integer

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

HW4 Solutions - CSE 20 Discrete Mathematics Spring 2010...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online