ILKL

1. CREATED BY K. VICTOR BABU
DATA STRUCTURE
23MT1102
Topic:
Direct Proofs
Department of
Mathematics
Session - 1

2. CREATED BY K. VICTOR BABU
AIM OF THE SESSION
To familiarize students with the basic concept of direct proofs
INSTRUCTIONAL OBJECTIVES
This Session is designed to:
To understand the concept of direct proofs.
To solve the problems on direct proofs.
LEARNING OUTCOMES
At the end of this session, you should be able to:
1. Able to Prove using Direct Proof method.
2. Able to understand proof methods .

3. CREATED BY K. VICTOR BABU
SESSION INTRODUCTION
Direct proof is a powerful method of mathematical reasoning that is used to prove theorems, propositions, and other
mathematical statements. It is a method of proof that is based on the laws of logic and involves starting with the
premises and using logical reasoning to arrive at the conclusion.
Direct proof is a fundamental tool in mathematics and has many applications in various fields such as physics,
engineering, computer science, and finance. For example, direct proof can be used to prove theorems in algebra,
geometry, calculus, and number theory. It is also used in cryptography to prove the security of encryption schemes.
In algebra, direct proof is used to prove many basic properties of numbers, such as the properties of addition and
multiplication. For example, direct proof can be used to prove that the product of two negative numbers is a positive
number, or that the sum of two odd numbers is an even number.
In geometry, direct proof is used to prove theorems about shapes and figures. For example, direct proof can be used to
prove that the sum of the interior angles of a triangle is 180 degrees, or that the Pythagorean theorem is true.

4. CREATED BY K. VICTOR BABU
SESSION INTRODUCTION
In calculus, direct proof is used to prove theorems about functions and their derivatives. For example, direct proof can
be used to prove the mean value theorem or the intermediate value theorem.
In number theory, direct proof is used to prove theorems about integers and their properties. For example, direct proof
can be used to prove that every integer greater than 1 is either prime or can be factored into primes.
Overall, direct proof is an essential tool for mathematicians and has many applications in various fields. By mastering
direct proof, mathematicians can prove theorems and propositions with confidence and apply their knowledge to
solve complex problems in real-world applications.

5. CREATED BY K. VICTOR BABU
SESSION DESCRIPTION
A proof is a sequence of statements. These statements come in two forms: givens and deductions. The
following are the most important types of "givens.''
Hypotheses: Usually the theorem we are trying to prove is of the form
𝑃1 ∧ ⋯ ∧ 𝑃𝑛 ⇒ 𝑄.
The P’s are the hypotheses of the theorem. We can assume that the hypotheses are true, because if one of
the Pi is false, then the implication is true.
Known results: In addition to any stated hypotheses, it is always valid in a proof to write down a theorem
that has already been established, or an unstated hypothesis (which is usually understood from context).
Definitions: If a term is defined by some formula it is always legitimate in a proof to replace the term by the
formula or the formula by the term.
We turn now to the most important ways a statement can appear as a consequence of (or deduction from)
other statements:

6. CREATED BY K. VICTOR BABU
SESSION DESCRIPTION (Cont..)
Tautology: If P is a statement in a proof and Q is logically equivalent to P, we can then write down Q.
Modus Ponens: If the formula P has occurred in a proof and 𝑃 ⇒ 𝑄 is a theorem or an earlier
statement in the proof, we can write down the formula Q. Modus ponens is used frequently, though
sometimes in a disguised form; for example, most algebraic manipulations are examples of modus
ponens.
Specialization: If we know "∀𝑥𝑃(𝑥),'' then we can write down "P(𝑥0)'' whenever 𝑥0 is a particular
value. Similarly, if "P(𝑥0)'' has appeared in a proof, it is valid to continue with "∃𝑥𝑃(𝑥)∃)''. Frequently,
choosing a useful special case of a general proposition is the key step in an argument.
When you read or write a proof you should always be very clear exactly why each statement is valid.
You should always be able to identify how it follows from earlier statements.

7. CREATED BY K. VICTOR BABU
SESSION DESCRIPTION (Cont..)
A direct proof is a sequence of statements which are either givens or deductions from
previous statements, and whose last statement is the conclusion to be proved.
Or
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given
statement by a straightforward combination of established facts, usually axioms,
existing lemmas and theorems, without making any further assumptions.
Example 1
The sum of two even integers equals an even integer
Consider two even integers x and y. Since they are even, they can be written as
𝑥 = 2𝑎, 𝑦 = 2𝑏, where a and b are integers
x +y = 2a+ 2b where , a and b are all integers.
It follows that x + y has 2 as a factor and therefore is even, so the sum of any two even integers is even.

8. CREATED BY K. VICTOR BABU
Class room delivery problems
If n is an even integer then 7n + 4 is an even integer.
2. If m is an even integer and n is an odd integer then m + n is an odd integer.
3. If m is an even integer and n is an odd integer then mn is an even integer.
4. If a, b and c are integers such that a divides b and b divides c then a divides c.

9. CREATED BY K. VICTOR BABU
SUMMARY
In this session, we introduce the concept of direct proofs,and studies its concept and solved it with
examples.

10. CREATED BY K. VICTOR BABU
SELF-ASSESSMENT QUESTIONS
1. Which of the following is not a step in a direct proof?
a) start with the hypothesis or premises that are given.
b) Use logical reasoning to deduce new information or properties from the premises.
c) Use counterexamples to disprove the conclusion.
d) Keep going until you arrive at the conclusion you want to prove …
…
2. What is the purpose of a direct proof?
a) To disprove a statement or proposition.
b) To prove a statement or proposition by starting with the premises and using logical steps to arrive at the
conclusion.
c) To prove a statement or proposition by contradiction.
d) To prove a statement or proposition using induction

11. CREATED BY K. VICTOR BABU
REFERENCES FOR FURTHER LEARNING OF THE SESSION
Reference Books:
1. Books:1.Kenneth H. Rosen, Discrete mathematics and its applications, McGraw Hill Publication, 2022.
2.Bernard Kolman, Robert Busby, Sharon C. Ross, Discrete Mathematical Structures, Sixth Edition Pearson
Publications, 2015
Reference Books:
1.Joe L Mott, Abraham Kandel, Theodore P Baker, Discrete Mathematics for Computer Scientists and
Mathematicians, Printice Hall of India, Second Edition, 2008.
2. Tremblay J P and Manohar R, Discrete Mathematical Structures with Applications to Computer Science, Tata
McGraw Hill publishers, 1st edition, 2001,India.
Sites and Web links:
1. https://www.youtube.com/watch?v=0gflLmuhHOg&list=PL0862D1A947252D20&index=7
2. https://www.youtube.com/watch?v=YFZzLQN5qOU
3. https://www.youtube.com/watch?v=ODqMYbTeWcU&t=55s
4. https://www.youtube.com/watch?v=0gflLmuhHOg&list=PL0862D1A947252D20&index=7

12. CREATED BY K. VICTOR BABU
THANK YOU
Team – Course Name