MATH 127: Exam 2 Review
Monday, March 14, 2011
1. Dene f : N Z by
f=
x/2
x is even
(x 1)/2 x is odd
1-to-1: Let m, n N and suppose f (m) = f (n). If m and n are both even, then m/2 = n/2
and thus m = n. If m and n are both odd then (m 1)/2 = (n 1)/2 and t
MATH 127: Exam 2 Review
Monday, March 14, 2011
1. Prove that |N| = |Z| by nding a bijection and proving that it is one.
2. Verify f : R3 R3 dened by f (x, y, z ) = (x + y, 2y + z, z x) is a bijection.
3. Verify that f : N N dened by f (n) = n2 is an injec
Concepts of Mathematics
Homework 10 Solutions
All of the following problems are graded this week.
Problem (10.5). Prove that every set of seven distinct integers contains a pair
whose sum or dierence is a multiple of 10.
Proof. Let a, b be two integers in
Concepts of Mathematics
Homework 9 Solutions
Problem (9.6). If A, B are independent, then Ac and B c are independent.
Proof. Assume A and B are independent. Then P (A B ) = P (A) P (B ). Also,
we have Ac B c = Ac (Ac B ) = Ac (B (B A). Hence
P (Ac B c ) =
Concepts of Mathematics
Homework 8 Solutions
Problem (7.18). Let p be an odd prime. Determine all solutions to 2n2 + n 0
mod p.
Proof. The solutions are n 0 mod p and n (p 1)/2 mod p. We have
2n2 + n 0 mod p i p|2n2 + n i p | n or p | (2n + 1) since p is