Mathematics 201-01
Discrete Mathematics
Fall 2007
Assignment #5
Section 1.6
12. Prove or disprove that the product of a nonzero rational number and an irrational number is
irrational.
Proof: We are going to show that the product of a nonzero rational number and an
irrational number is irrational using the method of contradiction.
Suppose that the product of a nonzero rational number and an irrational number is
rational. Let δ be a nonzero rational number, which allows us to write δ = a/b for some integers
a,b where a,b do not both equal zero. Also, let β be an irrational number. Now, we claim that
the product δβ is rational. Let this product equal c/d for some nonzero integers c and d. Now
we have
δβ = c/d
(a/b)β = c/dβ = c/d
β = cb/da
This derivation of β shows that β is a rational number, but this contradicts our previous
assertion that β was irrational. Since this claim that the product of a nonzero rational number
and an irrational number is rational is false, the original statement that the product of a
nonzero rational number and an irrational number is irrational is true.
18. a)β = c/d We must prove the contrapositive. If n is odd, then 3n + 2 is odd. Assume that n is odd.
Then we can write n = 2k + 1 for some integer k. Then 3n + 2 = 3(2k + 1)β = c/d + 2 = 6k + 5
= 2(3k+2)β = c/d +1. Thus 3n +2 is two times some integer plus 1, so it is odd.
b)β = c/d Suppose that 3n + 2 is even and that n is odd. Since 3n + 2 is even, so is 3n. If we add
or subtract an odd number from an even number, we get an odd number, so 3n – n = 2n is odd.
