COMP232 - Mathematics for Computer Science
Tutorial 8
Ali Moallemi
moa ali@encs.concordia.ca
Iraj Hedayati
h iraj@encs.concordia.ca
Concordia University, Winter 2016
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
1 / 12
Table of Contents
1
2.3 Functions
Exercise 53
Exercise 70
Exercise 71
Exercise 72
Exercise 73
Exercise 77
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
2 / 12
Exercise 53
Prove that if n is an integer, then bn/2c = n/2 if n is even and (n − 1)/2
if n is odd.
Answer: Let n be even, then there exists an integer k such that, n = 2k.
As a result we have,
bn/2c = b2k/2c = bkc = k = n/2.
Now let n be odd, then there exists an integer k such that, n = 2k + 1.
As a result we have,
bn/2c = b(2k + 1)/2c = bk + 1/2c = k = (n − 1)/2.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
3 / 12
Exercise 70
Suppose that f is an invertible function from Y to Z and g is an invertible
function from X to Y . Show that the inverse of the composition f ◦ g is
given by (f ◦ g)−1 = g −1 ◦ f −1
Answer:
∀x ∈ X x ∈ (f ◦ g)−1 (Z) ↔ ∃z ∈ Z; f ◦ g(x) = z ⇔
∀x ∈ X x ∈ (f ◦ g)−1 (Z) ↔ ∃z ∈ Z, ∃y ∈ Y ; f (y) = z ∧ g(x) = y ⇔
∀x ∈ X x ∈ (f ◦ g)−1 (Z) ↔ ∃z ∈ Z, ∃y ∈ Y ; f −1 (z) = y ∧ g −1 (y) = x ⇔
∀x ∈ X x ∈ (f ◦ g)−1 (Z) ↔ ∃z ∈ Z; g −1 ◦ f −1 (z) = x
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
4 / 12
Exercise 71
Let S be a subset of a universal set U . The characteristic function fS of S
is the function from U to the set {0, 1} such that fS (x) = 1 if x belongs
to S and fS (x) = 0 if x does not belong to S. Let A and B be sets.
Show that for all x ∈ U ,
a) fA∩B (x) = fA (x) · fB (x)
Answer:
fA∩B (x) = 1 ↔ x ∈ A ∩ B ↔ x ∈ A ∧ x ∈ B ↔
fA (x) = 1 ∧ fB (x) = 1 ↔ fA (x) · fB (x) = 1
b) fA∪B (x) = fA (x) + fB (x) − fA (x) · fB (x)
Answer:
fA∪B (x) = 1 ↔ x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B ↔
fA (x) = 1 ∨ fB (x) = 1 ↔ fA (x) + fB (x) − fA (x) · fB (x) = 1
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
5 / 12
Exercise 71 Cont...
Let S be a subset of a universal set U . The characteristic function fS of S
is the function from U to the set {0, 1} such that fS (x) = 1 if x belongs
to S and fS (x) = 0 if x does not belong to S. Let A and B be sets.
Show that for all x ∈ U ,
c) fA (x) = 1 − fA (x)
Answer:
fA (x) = 1 ↔ x ∈ A ↔ x ∈ U − A ↔
fA (x) = fU (x) − fA (x) = 1 − fA (x)
d) fA⊕B (x) = fA (x) + fB (x) − 2fA (x) · fB (x)
Answer:
fA⊕B (x) = 1 ↔ x ∈ A ⊕ B ↔ x ∈ (A ∪ B) − (A ∩ B) ↔
fA⊕B (x) = fA∪B (x) − fA∩B (x) =
From (a) and (b)
fA (x) + fB (x) − fA (x) · fB (x) − fA (x) · fB (x) =
fA (x) + fB (x) − 2fA (x) · fB (x)
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
6 / 12
Exercise 72
Suppose that f is a function from A to B, where A and B are finite sets
with |A| = |B|. Show that f is one-to-one if and only if it is onto
Answer: Let f be one-to-one, then |f (A)| = |A|. since |B| = |A| and
|f (A)| = |A|, then |f (A)| = |B|, and we know f (A) ⊆ B, consequently,
f (A) = B, which means f is onto.
Now let f be onto, as a result f (A) = B and |f (A)| = |B|, which implies
|f (A)| = |A|. So, we can imply that f is one-to-one.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
7 / 12
Exercise 73
Prove or disprove each of these statements about the floor and ceiling
functions.
a) dbxce = bxc for all real numbers x.
Answer: Since, bxc is an integer, so, dbxce = bxc holds.
b) b2xc = 2bxc whenever x is a real number.
Answer: If x = 0.6 then b2xc = b1.2c = 1, however
2bxc = 2b0.6c = 2 · 0 = 0
c) dxe + dye − dx + ye = 0 or 1 whenever x and y are real numbers.
Answer: If x or y is integer, by property 4.b in table1,
dxe + dye − dx + ye = 0. Now if neither is integer, so we there are
m, n ∈ Z and α, β ∈ R; 0 < α, β < 1, where x = m + α, y = n + β.
which means, m + n < x + y < m + n + 2. So, dx + ye = m + n + 1
or dx + ye = m + n + 2. Finally we have either,
dxe + dye − dx + ye = m + 1 + n + 1 − (m + n + 1) or
dxe + dye − dx + ye = m + 1 + n + 1 − (m + n + 2) which is equal to
0 or 1.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
8 / 12
Exercise 73
Prove or disprove each of these statements about the floor and ceiling
functions.
d) dxye = dxedye for real numbers x and y.
Answer: Let x = 2 and y = 0.1, Then we have,
dxye = d2 × 0.1e = 1, however, dxedye = d2ed0.1e = 2
e) d x2 e = b x+1
2 c for all real numbers x.
Answer: Let x = 0.5 then, d x2 e = 1, however, b x+1
2 c=0
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
9 / 12
Exercise 77
For each of these partial functions, determine its domain, codomain,
domain of definition, and the set of values for which it is undefined. Also,
determine whether it is a total function.
a) f : Z → R, f (n) = 1/n
Answer:
Domain: Z.
Codomain: R.
Domain of definition: Set of nonzero integers.
Set of undefined values:{0}.
Total function: No.
b) f : Z → Z, f (n) = dn/2e
Answer:
Domain: Z.
Codomain: Z.
Domain of definition: Z.
Set of undefined values:∅
Total function: Yes.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
10 / 12
Exercise 77 Cont...
For each of these partial functions, determine its domain, codomain,
domain of definition, and the set of values for which it is undefined. Also,
determine whether it is a total function.
c) f : Z × Z → Q, f (m, n) = m/n
Answer:
Domain: Z × Z.
Codomain: Q.
Domain of definition: Z × (Z − {0}).
Set of undefined values: Z × {0}
Total function: No.
d) f : Z × Z → Z, f (m, n) = mn
Answer:
Domain: Z × Z.
Codomain: Z.
Domain of definition: Z × Z.
Set of undefined values: ∅.
Total function: Yes
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
11 / 12
Exercise 77 Cont...
For each of these partial functions, determine its domain, codomain,
domain of definition, and the set of values for which it is undefined. Also,
determine whether it is a total function.
e) f : Z × Z → Z, f (m, n) = m − n if m > n.
Answer:
Domain: Z × Z.
Codomain: Z.
Domain of definition: {(m, n) ∈ Z × Z | m > n}
Set of undefined values: {(m, n) ∈ Z × Z | m ≤ n}
Total function: No.
Ali Moallemi, Iraj Hedayati
COMP232 - Mathematics for Computer Science
12 / 12