Summary: Math 3130, Abstract Algebra
Homework 2 Key
3.5Are the two sets "the positive multiples of 3" and "the positive multiples of 5" the same size?
Solution: Yes. Proof: Recall that two sets are the same size if there is a bijection between them. The
map f(n) = 3n and g(n) = 5n are both bijections of the positive integers with, respectively, the positive
multiples of 3 and 5. To see this note that if two positive multiples of 3 (or 5) are the same they must
be multiples of the same n yielding injectivity. Any positive multiple of 3 or 5 is a multiple of some
positive integer yielding surjectivity. Since both sets have a bijection with the positive integers, we see
they are both the same size as the positive integers and so the same size as one another. 2
3.12Suppose that 2n
- 1 is prime. Prove that n is prime.
Solution: Suppose n is not prime and that n = ab for 1 < a b < n. Notice that by using the sum
formula for finite geometric series we get:
- 1)(1 + 2a
+ · · · + 2(b-1)a