epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

104 The Nuts and Bolts of Proofs
4. (a) Are the following two sets equal?
A = {all integer numbers that are multiples of 15}
B = {all integer numbers that are multiples of 3 and 5}
(b) Are the following two sets equal?
^=: {all integer numbers that are multiples of 15}
B = {all integer numbers that are either multiples of 3 or multiples
of 5}
5. Let a and b be two real numbers with a^O. The solution of the equation
ax = b exists and is unique.
6. The counting number n is odd if and only if n^ is odd.
7. Let a and b be two real numbers. The following statements are equivalent:
(a) a<b and a>b
(b) a-b = 0.
8. Every nonzero real number has a unique reciprocal.
9. Let p, q, and n be three positive integers. If p and q have no common
factors, then q does not divide p".
10. For every integer n > 0
11. \/2 is an irrational number.
11 11 11 11 n
12 23 34 nn+1 n+1
12. Prove algebraically that two distinct Unes have at most one point in
common.
13. All negative numbers have negative reciprocals. (See Exercise 8.)
f 3n + 2l^
14. Let I Un = } . Then lim a„ = 3.
I " Jn=l
15. The remainder of the division of a polynomial P{x) by the monomial
x — ais the number P(a).
16. Let P(x) be a polynomial of degree larger than or equal to 1. The
following statements are equivalent:
(1) The number x = ais a, root of P{x).
(2) The polynomial P(x) can be exactly divided by the monomial x-a.
(3) The monomial x — a is a factor of the polynomial P(x).
17. Let/be a differentiable function at the point x = a. Then/is continuous
at that point. (*)
18. All prime numbers larger than two are odd.