Chapter 1_Sets. Operation sets, Principle of inclusion and exclusion etc.

DISCRETE
MATHEMATICS
Chapter 1: Sets

Course Description
Nowadays, information technology has profoundly changed
the way we live and the way we think. Problems arising in the
entire spectrum of information technology have an increasing
influence on mathematics, and especially on discrete
mathematics. Basically, discrete mathematics is the branch of
mathematics that studies the underlying principles which
govern discrete structures and the binary universe. Such
principles are essential and effective in the implementation of
algorithms, performance analysis, and information
management. To build a sound scientific foundation for the
information age requires collective interdisciplinary efforts, to
which discrete mathematicians can contribute in numerous
ways. This course emphasizes not only the fundamentals of
discrete mathematics and its advanced topics, but also how to
express ideas in clear mathematical language. Students will
discover the importance of discrete mathematics in describing
structures and problem solving but will also develop reasoning
skills that will continue to be useful throughout their careers.

AT THE END
OF THIS
LESSON:
1.1 Understand the fundamental concepts of sets,
including elements and the language used to describe
collections.
1.2 Learn and apply common operations such as
unions, intersections, complements, and Cartesian
products.
1.3 Master the technique for counting the number of
elements in the union of overlapping sets, crucial for
solving complex counting problems.
1.4 Grasp the Principle of Mathematical Induction as
a method for proving statements about integers and
sequences.
1.5 Explore the Strong Form of Mathematical
Induction, which extends the principles of induction to
more complex scenarios.

Content
1.1 Basic Definitions
1.2 Operations on Sets
1.3 The Principle of Inclusion-Exclusion
1.4 Mathematical Induction

Activity 2
1. List the members of this sets:
a. {x/x is a positive integer less than 12}
b. {x/x is a nonnegative integer, a perfect square and less than 100}.
2. For each of these pair sets, determine whether the first set is a subset of the second set,
and indicate not a subset if it is not.
a.The set of fruits, the set of citrus fruits
b. The set of students studying discrete math, the set of students studying data structure.
3. Use the venn diagram to illustrate all months of the year whose names do not contain
the letter R.
4. Let A= {a,b,c}, B={x,y}, C={0,1}. Find C X A X B
5. How would you assess the difficulty of the first topic. Please give 2-3 reasons.

Chapter 1_Sets. Operation sets, Principle of inclusion and exclusion etc.

Activity 3
1. Prove that
2. The bit strings for the sets {1, 2, 3, 4, 5} and {1, 3, 5, 7,
9} are 11 1110 0000 and 10 1010 1010, respectively. Use
bit strings to find the union and intersection of these sets.
3. Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find A
∪ B, A ∩ B, A − B, and B − A.
4. Show that if A, B, and C are sets, then A ∩ B ∩ C = A ∪ B
∪ C
a) using a membership table.
5.What are the importance of this topics and how would you
apply it as an IT student?

Content
1.1 Basic Definitions
1.2 Operations on Sets
1.3 The Principle of
Inclusion-Exclusion
1.4 Mathematical Induction

THE PRINCIPLE OF
INCLUSION AND
EXCLUSION
(FOR THREE SETS)
By
Sharvari navgire

Definition & Formula
|A∪B∪C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|
The Principle of Inclusion and Exclusion for three sets is given by this formula

Example:
If A = { 1, 2 , 3 }
B = { 2, 3 }
C = { 3, 4, 5 }
|A| = 3
|B| = 2
|C| = 3
A ∩ B = { 2, 3 }
|A ∩ B| = 2
A ∩ C = { 3 }
| A ∩ C| = 1
B ∩ C = { 3 }
|B ∩ C| = 1
A ∩ B ∩ C = { 3 }
|A ∩ B ∩ C | = 1
A ∪ B ∪ C = { 1, 2, 3, 4, 5 }
|A ∪ B ∪ C | = 5
|A ∪ B ∪ C | = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C |
5 = 3 + 2 + 3 – 2 – 1 – 1 + 1
5 = 5

Try this!
Exercise 1
Exercise 2

MATHEMATICAL INDUCTION
Mathematical induction is a powerful and fundamental technique for
proving results about all natural numbers.
Ø It is most important when it is possible to write down a proof for each
individual natural number but difficult-or even impossible-to give a single
direct proof that works for all natural numbers.
Ø This proof technique also often is used to prove that algorithms are correct
and to determine expressions for the complexity of algorithms.
In general, mathematical induction can be used to prove statements that
assert that P(n) is true for all positive integers n, where P(n) is a propositional
function. A proof by mathematical induction has two parts, a basis step,
where we show that P(1) is true, and an inductive step,where we show that
for all positive integers k, if P(k) is true, then P(k + 1) is true.

MATHEMATICAL INDUCTION
PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all
positive integers n, where P(n) is a propositional function, we complete two
steps:
BASIS STEP: We verify that P(1) is true.
INDUCTIVE STEP: We show that the conditional statement P(k) → P(k + 1) is
true for all positive integers k.

WAYS TO REMEMBER HOW MATHEMATICAL
INDUCTION WORKS

Example 1
Show that if n is a positive integer, then
1 + 2 + ⋯ + n = n(n + 1)/2.
Solution:
Let P(n) be the proposition that the sum of the first n positive integers,
1 + 2 + ⋯ n = n(n+1)/2 , is n(n + 1)∕2.
We must do two things to prove that P(n) is true for n = 1, 2, 3, … .
Namely, we must show that P(1) is true and that the conditional statement P(k)
implies P(k + 1) is true for k = 1, 2, 3, … .

Example 1
1 + 2 + ⋯ + n = n(n + 1)/2.
Solution:
BASIS STEP:
P(1) is true, because 1 = 1(1 + 1)/ 2 . (The left-hand side of this equation is 1
because 1 is the sum of the first positive
integer. The right-hand side is found by
substituting 1 for n in n(n + 1)∕2.)

Example 1
1 + 2 + ⋯ + n = n(n + 1)/2.
Solution:
INDUCTIVE STEP:
For the inductive hypothesis we assume that P(k) holds for an arbitrary positive
integer k. That is, we assume that
1 + 2 + ⋯ + k = k(k + 1)/2 .
Under this assumption, it must be shown that P(k + 1) is true, namely, that
1 + 2 + ⋯ + k + (k + 1) = (k + 1)[(k + 1) + 1]/2 = (k + 1)(k + 2)/2 is also true.

Example 1
1 + 2 + ⋯ + n = n(n + 1)/2.
Solution:
INDUCTIVE STEP:
We now look ahead to see how we might be able to prove that P(k + 1) holds
under the assumption that P(k) is true. We observe that the summation in the left-
hand side of P(k + 1) is k + 1 more than the summation in the left-hand side of
P(k). Our strategy will be to add k + 1 to both sides of the equation in P(k) and
simplify the result algebraically to complete the inductive step.

Example 1
1 + 2 + ⋯ + n = n(n + 1)/2.
Solution:
INDUCTIVE STEP:
We now return to the proof of the inductive step. When we add k + 1 to both
sides of the
equation in P(k), we obtain
1 + 2 + ⋯ + k + (k + 1) = k(k + 1)/2 + (k + 1)
= k(k + 1) + 2(k + 1)/2
= (k + 1)(k + 2)/2
This last equation shows that P(k + 1) is true under the assumption that P(k) is true.
This completes the inductive step.

Example 1
1 + 2 + ⋯ + n = n(n + 1)/2.
Solution:
We have completed the basis step and the inductive step, so by mathematical
induction we know that P(n) is true for all positive integers n. That is, we have
proven that 1 + 2 + ⋯ + n = n(n + 1)∕2 for all positive integers n.

Problem Set 1
1. A computer company receives 350 applications from college graduates for a job
planning a line of new web servers. Suppose that 220 of these applicants majored in
computer science, 147 majored in business, and 51 majored both in computer science
and in business. How many of these applicants majored neither in computer science nor
in business?
2. Use the principle of inclusion–exclusion to find the number of positive integers less than
1,000,000 that are not divisible by either 4 or by 6.
3. . How many natural numbers between 1 and 30,000,000 (including 1 and 30,000,000)
are divisible by 2, 3, or 5?
4. In the game of tic-tac-toe, every game ends with one player winning or with a draw.
In a tic-tac-toe tournament, the players merely count the number of times they win
or draw. The match winner is the player with the larger total. If a match between two
players A and B consists of 25 games, player A has a score of 19, and player B has a
score of 23, how many draws were there?
5. Use mathematical induction to show that
1 + 2 + 22 + ⋯ + 2n = 2n+1 − 1
for all nonnegative integers n.

PLEASE PREPARE
FOR OUR FIRST
LONG QUIZ
TOPIC: CHAPTER 1

Chapter 1_Sets. Operation sets, Principle of inclusion and exclusion etc.

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