Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

TMF1814 DISCRETE MATHEMATICS

TUTORIAL 2

S1 2015/2016

TMF1814 DISCRETE MATHEMATICS
TUTORIAL 2
[ANSWERS]
1
...
Find these values
...

(a) ⌊1
...
1⌉ = 2
(c) ⌈2
...
99⌉ = -2
(e) ⌊½ + ⌈½⌉⌋ = ⌊½ + 1⌋ = ⌊3⁄2⌋ = 1
(f) ⌈⌊½⌋ + ⌈½⌉ + ½⌉ = ⌈0 + 1 + ½⌉ = ⌈3⁄2⌉ = 2

2
...

(b) 𝑓( 𝑚, 𝑛) = 𝑚² − 𝑛²
This is not onto since for example, 2 is not in the range
...
In either
case, both m – n and m + n are then even, so this expression is divisible by 4 and
hence cannot equal 2
...
To achieve negative values, we set m = 0 and to achieve nonnegative
values we set n = 0
...
in fact, the range here is clearly a
subset or the range in that part
...
Determine whether each of these functions is a bijection from R to R
...
Otherwise we must explain why the
function is not one-to-one or not onto
...

It is also not onto since the range is the interval (-∞, 7]
...


(
)
(c) 𝑓 ( 𝑥 ) = 𝑥 + 1 ⁄ 𝑥 + 2)
(
This function is a bijection
...
The inverse is 𝑓 −1 ( 𝑥 ) = √𝑥 − 1

4
...
Find 𝑓( 𝑆) if
In all parts, we simply need to compute the values 𝑓(−1), 𝑓(0), 𝑓(2), 𝑓(4), 𝑓(7) and
collect the values into a set
...
Find 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓, where 𝑓( 𝑥) = 𝑥² + 1 and 𝑔( 𝑥 ) = 𝑥 + 2, are functions from R to R
...
4 Sequences and Summations

2

TMF1814 DISCRETE MATHEMATICS

TUTORIAL 2

S1 2015/2016

6
...

(a) 2n-1 = 28-1 = 128
(b) 7 = 7
(c) 1 + (−1) 𝑛 = 1 + (−1)8 = 0
(d) – (−2) 𝑛 = – (−2)8 = -256

7
...
Assuming that your formula
or rule is correct, determine the next three terms of the sequence
...
What are the values of these sums, where S = {1, 3, 5, 7}?
(a) ∑ 𝑗∈𝑆 𝑗
1 + 3 + 5 + 7 = 16

(b) ∑ 𝑗∈𝑆 𝑗 ²
1² + 3² + 5² + 7² = 84
(c) ∑ 𝑗∈𝑆 1⁄ 𝑗
(1/1) + (1/3) + (1/5) + (1/7) = 176/105
(d) ∑ 𝑗∈𝑆 1
1+1+1+1= 4

9
...

(a) ∑3 ∑2 (3𝑖 + 2𝑗)
𝑖=0 𝑗=0
(0 + 0) + (0 + 2) + (0 + 4) + (3 + 0) + (3 + 2) + (3 + 4) + (6 + 0) + (6 + 2) + (6 + 4)
+ (9 + 0) + (9 + 2) + (9 + 4) = 78

(b) ∑3 ∑2 𝑗
𝑖=1 𝑗=0
(0 + 1 + 2) + (0 + 1 + 2) + (0 + 1 + 2) = 9

(c) ∑2 ∑3 𝑖²𝑗³)
𝑖=0 𝑗=0
(0 + 0 + 0 + 0) + (0 + 1 + 8 + 27) + (0 + 4 + 32 + 108) = 180

2
...
There are four major auto routes from Boston to Detroit and six from Detroit to Los
Angeles
...

11
...
How many bit strings are there of length of 6 or less?
We use the sum rule, adding the number of bit strings of each length up to 6
...
1)

5



---