Description

Worksheet 5 [pdf, docx] covers Sections 2.3 (part 2), 4.2, and 2.4. Only the subsections in the worksheet are required for the course.

Assignment:

Solve all the 12 exercises on the worksheet on your own paper or by using your favorite software.

Submit your solutions here.

The total of 12 exercises will be graded by completion. On top of that, I will consider 3 among the 12 exercises for correctness. MT 210

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Math 210, Concepts from Discrete Mathematics

Worksheet 5

This is a video guideline. Complete all the proposed exercises after watching the videos posted

on D2L.

Section 2.3 part 2

Inverse Functions and Compositions of Functions.

Watch the videos about inverse and composition of functions.

Exercise 1. Read Definitions 9 and 10 and write the definitions of inverse function and

composition of functions.

Exercise 2. Let f: Z → Z be such that f(n) =n+ 2. Is f invertible and if it is, what is its inverse?

Exercise 3. Determine f ◦ g and g ◦ f where f(x) = x2 + 1 and g(x) = x+ 2 are functions from R to R.

Exercise 4. Let f: R × R → R be such that f(x, y) = x+y. Is f invertible and if it is, what is its inverse?

Can we restrict the domain or the codomain of the function f so that f becomes invertible?

What is the inverse in this case?

Section 4.2

Representation of Integers.

Watch the videos about representation of integers.

Exercise 5. Convert each of the following integers into the required expansion. Show some of

your computations.

(a) 321 into binary

(b) 1023 into octal

(c) (2653)7 into decimal

(d) 456 into ternary

(e) 56291 into hexadecimal

(f) (11110111)2 into base 7

(g) (126727)8 into binary

(h) (10101101)2 into hexadecimal

Algorithms for Integer Operations.

Watch the videos about integer operations.

Exercise 6. Perform the following sums and products. Show your computations.

(a) (110)2 · (101)2

(b) (10110101)2 + (11000111)2

(c) (2112)3 + (12021)3

(d) (112)3 · (210)3

(e) (23)6 · (14)6

(f) (234)6 + (421)6

(g) (1AE)16 ·(BBC)16

(h) (20CBA)16 + (A01)16

Section 2.4

Sequences

Watch the videos about sequences.

Exercise 7.

List the first 10 terms of the sequence an = floor(√n) for n ≥ 1 where floor(x) is the largest

integer less than or equal to the real number x.

Exercise 8. List the first 10 terms of the sequence an = 2an-1 − an-2, with a0 = 0, a1= 1.

Recurrence Relations

Watch the videos about recurrence relations.

Exercise 9. Assume that the population of the world in 2010 was 6.9 billion and is growing at

the rate of 1.1% a year. Set up a recurrence relation for the population of the world n years

after 2010.

Exercise 10. Show that the sequence an= 2n is a solution of the recurrence relation

an= 5an-1 − 6an-2

Summations

Watch the videos about summations.

Exercise 11. Find the value of the summation

∑ j from 0 to 3 12 · (−3)j

Exercise 12. Find the value of the following sum as a closed formula in terms of n.

∑j from 0 to n (2·3j+ 3·2j)

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sequences

discrete mathematics

recurrence relations

one to one function

Representation of Integers

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