MATH 235 01: Basic Algebra Fall 2002

Assignment 1

1) Prove that 5 is a factor of 2^{4n−2} + 1 for every positive integer n.

2) Prove that 2^{n} > n^{2} for every integer n ≥ 5.

3) Let A be a set of n elements (n is a natural number). Prove that A has precisely 2^{n} different subsets.

4) Let R be the set of all real numbers. Prove that the function f : R → R given by f(x) = −2x + 3 is a bijection.

5) Find a bijection f : N → Z where N is the set of natural numbers and Z is the set of integers.

6) Let A, B be subsets of a set U. Prove that U − (A ∪ B) = (U − A) ∩ (U − B)