Problem Sheet 3
MATH 363
Winter 2017
On this sheet all exercises are assessed and solutions are due before 5pm on Thursday 16
February 2017.
1. Let x be a real number and recall that bxc and dxe denote the floor and ceiling of x
respectively.
(a) Prove that the following three statements are equivalent:
i. bxc = n
ii. n ≤ x < n + 1
iii. x − 1 < n ≤ x.
(b) Prove that the following three statements are equivalent:
i. dxe = n
ii. n − 1 < x ≤ n
iii. x ≤ n < x + 1.
(c) Show that x − 1 < bxc ≤ x ≤ dxe < x + 1.
(d) Show that d−xe = − bxc and that b−xc = − dxe.
2. Let x > 0. Show that bxc + x + 13 + x + 23 = b3xc. (Hint: write x = n + where
n ∈ N0 and ∈ [0, 1). Then distinguish whether < 13 or ∈ [ 13 , 32 ) or ∈ [ 23 , 1).)
3. Let f : [0, 1] ∪ [2, 3] −→ R be defined by f (x) = x3 .
(a) Show that f is injective.
(b) Show that f is not surjective.
(c) Find a subset A of R such that the map g : [0, 1] ∪ [2, 3] −→ A is a bijection.
4. Show that for all n ∈ Z+ we have
n
X
(2k − 1) = n2 .
(1)
k=1
(Recall that Z+ denotes the set of all strictly positive integers, i.e. Z+ = {1, 2, 3, . . .}.)
5. Prove by induction that
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(a) 32n − 1 is divisible by 8
(b) 11n+1 + 122n−1 is divisible by 19
for all n ∈ Z+ .
6. (a) Prove by induction that if n ∈ Z+ and A and B are sets with |A| = |B| = n then
there exists a bijection f : A −→ B.
(b) Conversely show that for n ∈ Z+ that if |A| = n and there exists a bijection f : A −→
B then |B| = n.
7. How many integers x lie strictly between 0 and 100 (i.e. 0 < x < 100) and are divisible
by 3 or divisible by 4? (As usual we interpret ‘or’ to be inclusive!)
8. A palindrome is a word that reads the same forwards and backwards (e.g. bib, noon,
level). For the purposes of this question our alphabet consists of the 26 lower-case letters
of the standard English alphabet.
(a) How many palindromes of length 3 are there?
(b) How many palindromes of length 4?
(c) Give a formula for the number of palindromes of length n, where n ∈ Z+ .
(To clarify: it is not required for these palindromes to make any sense in English, or any
other language! For example, xyx is a perfectly good palindrome of length 3.)
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