MA304/MA310, Discrete Mathematics, Week 3 exercises.
Recall: A real number r is called rational if it is a fraction r =
a and b (b 6= 0).
Exercise 1. Is
√
a
b
with integer numbers
5 a rational number? Give a proof of your answer.
Exercise 2. Let x, y be real numbers. Prove or disprove:
1. If x is rational and y is rational, then x + y is rational.
2. If x is rational and y is irrational, then x + y is irrational.
3. If x is rational and y is irrational, then x · y is irrational.
4. If x · y is rational, then x or y are rational.
5. If x + y is irrational, then x or y are irrational.
6. If x + y is rational, then x is irrational or y is rational.
Exercise 3. Let N = {1, 2, 3, · · ·} denote the set of natural numbers and let R be the
set of real numbers. True or false? Explain!
1. ∀x ∈ N ∃y ∈ R with y < x;
2. ∀x ∈ R ∃y ∈ N with y < x;
3. ∃x ∈ N ∃y ∈ N such that [(2x + y = 5) and (x − 3y = −8)];
4. ∃x ∈ N ∃y ∈ N such that [(3x − y = 7) and (2x + 4y = 3)].
Exercise 4. Prove the following statements using mathematical induction.
1. 13 + 23 + 33 + . . . + n3 = n2 (n + 1)2 /4 for n ≥ 1.
2. 2n > n2 for all n ≥ 5
Exercise 5. For which natural numbers n does the inequality 4n ≤ n2 − 7 hold?
Explain.
Exercise 6. Prove that for every natural number n, the number n5 − n is divisible by
5. Conclude that 110221500 − 110221100 is a multiple of 10.
Exercise 7. Prove that
∀n ∈ N : 1 · 3 + 2 · 4 + 3 · 5 + · · · + n(n + 2) =
n(n + 1)(2n + 7)
.
6
Exercise 8. Let s0 := a, ∀ n ≥ 1: sn = 2sn−1 + b. Find a formula for sn .
Exercise 9. Prove:
n
X
i=1
Exercise 10. Prove:
n
X
i=1
i2 =
n(n + 1)(2n + 1)
.
6
i(i + 1) =
n(n + 1)(n + 2)
.
3
Exercise 11. Prove: n straight lines, no two of which are parallel and no three of
which pass through one point, cut the plane into 12 (n2 + n + 2) regions.
Exercise 12. Compute:
1.
10
X
1
j=1
2.
k
X
`
j=1
3.
10
Y
j
j=1
4.
1237
Y
(ej − 1)
j=0
5.
6.
m
Y
k+1
k
k=1
n
X
j
j=m
7.
k
X
n=1
1
= 1/2 + 1/6 + 1/12 + · · ·
n(n + 1)