The LNM Institute of Information Technology
Jaipur, Rajasthan
MATH-I Assignment #1
(Real Number System)
Notation: Q : set of rational numbers, R : set of real numbers, N: set of natural
numbers, a|b : a divides b.
1. Show that
√
(a) p is not a rational for any prime p.
(b) Prove that there is no rational number whose square is n, where n is not a
perfect square.
√ √
(c) 3 2, 5 16 are not rational.
(Hint: Let p be a prime, and m and n be natural numbers. If p|mn , then p|m. )
2. Show that log10 2 is not rational.
a
(Hint: Take log10 2 = ab ∈ Q with gcd(a, b) = 1. Take 2 = 10 b ⇒ 2b = 10a ⇒ 2b =
b
2a 5a ⇒ 52a = 2a )
3. If r is rational (r 6= 0) and x is irrational, prove that r + x and rx are irrational.
4. Find the infimum and supremum (if exists) of the following sets:
(a) S1 = n2 : n ∈ N .
n
o
n
(b) S2 = (−1)
:
n
∈
N
.
n
(c) S3 = −1
:n∈N .
n2m
(d) S4 = m+n : m, n ∈ N .
5. Let S be a nonempty subset of R and β be a real number. Then show that β = inf S
if and only if
(a) β is a lower bound of S.
(b) For each > 0, there exists a x ∈ S such that β + > x.
6. Let A, B be nonempty and bounded subsets of R with A ⊂ B. Prove that
inf B ≤ inf A ≤ sup A ≤ sup B.
7. Show that the following two statements are equivalent.
(a) Every nonempty subset of R that is bounded above has the supremum in R.
(b) Every nonempty subset of R that is bounded below has the infimum in R.
8. Show that the following two statements are equivalent.
(a) Given any x ∈ R, there exists n ∈ N such that n > x.
(b) If a, b ∈ R and a > 0, then there is a positive integer n such that na > b.
\ 1
is an empty
9. Use the Archimedean property of real numbers to show that
0,
n
n∈N
set.