This is a presentation that deals with the theory of computation formal proofs and the different types of proofs that is used in theory of computation.

1. Theory of Computation

2. Introduction to formal proof
• A proof is a convincing logical argument that a
statement is true.
• A formal proof or derivation is a
finite sequence of sentences (called well-formed
formulas in the case of a formal language), each of
which is an axiom, an assumption, or follows from the
preceding sentences in the sequence by a rule of
inference
• Formal proof techniques are indispensable for proving
theorems in theory of computation.
• Understanding how correct programs work is
equivalent to proving theorems by induction.

3. Formal Proofs
There are 5 different forms of proofs
1. Deductive Proof
2. Proof by Induction(Inductive Proof)
3. Proof by Contradiction
4. Proof by Counter example
5. Proof by Contrapositive

4. Deductive Proof
• From the given statement(s) to a conclusion statement (what we want to prove)
i.e. Given a hypothesis H, and some statements, generate a conclusion C.
• Each step of a deductive proof MUST follow from a given fact or previous statements (or their
combinations) by an accepted logical principle.
• The theorem that is proved when we go from a hypothesis H to a conclusion C is the
statement ’’if H then C’’. We say that C is deduced from H.
“If H then C” i.e C is deduced from H

7. Proof by Contradiction
• Start with a statement contradictory to given
statement and then prove it leads to
contradiction
• Suppose that we want to prove H and we
know that it is true. Instead of proving H
directly, we may instead show that assuming
¬H leads to a contradiction. Therefore H must
be true.

8. Proof by Counter Example
• A counter example disproves a statement by
giving a situation where the statement is false.
• It is the technique where a statement is
shown to be wrong by finding a single
example whereby it is not satisfied.
• An example which disproves a proposition.
For example, the prime number 2 is
a counterexample to the statement "All prime
numbers are odd."

9. Proof by Contrapositive
• Proof by contrapositive takes advantage of the
logical equivalence between "H implies C" and
"Not C implies Not H".
• For example, the assertion "If it is my car, then
it is red" is equivalent to "If that car is not red,
then it is not mine".
• To prove "If P, Then Q" by the method of
contrapositive means to prove "If Not Q, Then
Not P".

10. The Principle of Mathematical
Induction

11. Mathematical Induction
• Suppose we have some statement P(n) and
we want to demonstrate that P(n) is true for
all n belonging to N.
• Even if we can provide proofs for P(0), P(1),
..., P(k), where k is some large number, we
have accomplished very little.
• However, there is a general method, the
Principle of Mathematical Induction

12. The principle of mathematical induction
is a tool which can be used to prove a
wide variety of mathematical statements.
Each such statement is assumed as P(n)
associated with positive integer n, for
which the correctness for the case n=1 is
examined.
Then assuming the truth of P(k) for some
positive integer k, the truth of P(k+1) is
established.

13. There is a given statement P(n) involving the natural
number n such that
(i) The statement is true for n=1, i.e., P(1) is true
This is called the proof for the basis.
(ii) If the statement is true for n=k (where k is some
positive integer ), then the statement is also true for
n=k+1
i.e., truth of P(k) implies the truth of P(k+1).
This is called the induction hypothesis.
(iii) Then, P(n) is true for all natural numbers n.
Proof by Induction…..

14. Example…
• Prove that 1 + 3 + 5 + ……+ n = n(n+1)/2
Solution
(a) Proof for the basis.
For n=1, LHS = 1 and RHS = 1(1+1)/2 =1
Hence the result is true for n = 1.
(b) By Induction hypothesis, we assume it is true
for n = k,
1 + 3 + 5 + ……+ k= k(k+1)/2
we have to prove that this is true for n = k+1
also
ie 1 + 3 + 5 + ……+ k + (k+1) = (k + 1)(k+2)/2

15. LHS = 1 + 3 + 5 + ……+ k + (k+1)
= k(k+1)/2 + (k+1)
= (k(k +1) + 2(k+1))/2
= ((k+1)(k+2))/2
=RHS

16. Examples…
Prove by induction
1. 1 + 4 + 7+…….+ (3n -2) =
𝑛(3𝑛−1)
2
2. 13 + 23 + …….. + n3 =
𝑛2 𝑛+1 2
4
3. 4+9+14+19+……+(5n-1)=n/2(3+5n)
4. -1+2+5+8+……+(3n-4)=n/2[3n-5]
5. 1/2+1/4+1/8+…. 1/2n=(2n-1)/2n
6. Prove n!>2n for n>=4

17. Examples….