Instructor: Sourav Chakrabortyneelraha/Resources/Sem 2/DMAT/class2.pdf · Negating a sentence What...

Discrete MathematicsLecture 2: Proof Technique

Instructor: Sourav Chakraborty

Discrete Mathematics Lecture 2: Proof Techniques

Propositional logic

Every statement (or proposition) is either TRUE orFALSE.

A statement can be formed using other statementsconnected to each other by 5 kinds of connectives: AND,OR, NOT, IMPLIES and IFF.

A statement can have unspecified terms, called variables.All the variables has to be properly quantified using thetwo quantifiers FORALL and THERE EXISTS.

Discrete Mathematics Lecture 2: Proof Techniques

Propositional logic

Every statement (or proposition) is either TRUE orFALSE.

A statement can be formed using other statementsconnected to each other by 5 kinds of connectives: AND,OR, NOT, IMPLIES and IFF.

A statement can have unspecified terms, called variables.All the variables has to be properly quantified using thetwo quantifiers FORALL and THERE EXISTS.

Discrete Mathematics Lecture 2: Proof Techniques

Propositional logic

Every statement (or proposition) is either TRUE orFALSE.

A statement can be formed using other statementsconnected to each other by 5 kinds of connectives: AND,OR, NOT, IMPLIES and IFF.

A statement can have unspecified terms, called variables.All the variables has to be properly quantified using thetwo quantifiers FORALL and THERE EXISTS.

Discrete Mathematics Lecture 2: Proof Techniques

Propositional logic

Every statement (or proposition) is either TRUE orFALSE.

A statement can be formed using other statementsconnected to each other by 5 kinds of connectives: AND,OR, NOT, IMPLIES and IFF.

A statement can have unspecified terms, called variables.All the variables has to be properly quantified using thetwo quantifiers FORALL and THERE EXISTS.

Discrete Mathematics Lecture 2: Proof Techniques

Checking correctness of a statement or theorem

Any meaningful statement / proposition / thoerem can bewriten as a mathematical logic statement.

A statement is consistent or correct if for any setting of theinput variables (smaller statements) to TRUE or FALSEthe statement always evaluate to TRUE.

Discrete Mathematics Lecture 2: Proof Techniques

Checking correctness of a statement or theorem

Any meaningful statement / proposition / thoerem can bewriten as a mathematical logic statement.

A statement is consistent or correct if for any setting of theinput variables (smaller statements) to TRUE or FALSEthe statement always evaluate to TRUE.

Discrete Mathematics Lecture 2: Proof Techniques

Checking correctness of a statement or theorem

Any meaningful statement / proposition / thoerem can bewriten as a mathematical logic statement.

A statement is consistent or correct if for any setting of theinput variables (smaller statements) to TRUE or FALSEthe statement always evaluate to TRUE.

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Is A =⇒ B is equivalent to (¬B ∧A) = FALSE

A B A =⇒ B (¬B ∧A) = F

F FF TT FT T

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Is A =⇒ B is equivalent to (¬B ∧A) = FALSE

A B A =⇒ B (¬B ∧A) = F

F FF TT FT T

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Is A =⇒ B is equivalent to (¬B ∧A) = FALSE

A B A =⇒ B (¬B ∧A) = F

F FF TT FT T

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Is A =⇒ B is equivalent to (¬B ∧A) = FALSE

A B A =⇒ B (¬B ∧A) = F

F FF TT FT T

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Is: A =⇒ B is equivalent to (¬B ∧A) = FALSE

A B A =⇒ B (¬B ∧A) = F

F F TF T TT F FT T T

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Is: A =⇒ B is equivalent to (¬B ∧A) = FALSE

A B A =⇒ B (¬B ∧A) = F

F F T TF T T TT F F FT T T T

Discrete Mathematics Lecture 2: Proof Techniques

Problems on Propositional Logic

Prove that (a =⇒ b) is equivalent to (¬a ∨ b).Prove that (a ⇐⇒ b) is equivalent to(a =⇒ b) ∧ (b =⇒ a).

Prove that (p ∨ q) =⇒ r is equivalent to(p =⇒ r) ∧ (q =⇒ r).

Prove that (a =⇒ b) is equivalent to (¬b =⇒ ¬a).

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Another approach is to use some already proved rules tosimplify the formulas before using the brute force truthtableapproach.

Discrete Mathematics Lecture 2: Proof Techniques

Checking Equivalence

Definition

Two statements are equivalent if their TRUTHTABLES are thesame.

Another approach is to use some already proved rules tosimplify the formulas before using the brute force truthtableapproach.

Discrete Mathematics Lecture 2: Proof Techniques

Rules of Propositional Logic

Let p, q and r be propositions.

1 Commutative law:

(p ∨ q) = (q ∨ p) and (p ∧ q) = (q ∧ p)2 Associative law:

(p ∨ (q ∨ r)) = ((p ∨ q) ∨ r) and (p ∧ (q ∧ r)) = ((p ∧ q) ∧ r)3 Distributive law:

(p ∨ (q ∧ r)) = (p ∨ q) ∧ (p ∨ r) and(p ∧ (q ∨ r)) = (p ∧ q) ∨ (p ∧ r)

4 De Morgan’s Law:

¬(p ∨ q) = (¬p ∧ ¬q) and ¬(p ∧ q) = (¬p ∨ ¬q)

Discrete Mathematics Lecture 2: Proof Techniques

Rules for Negation

¬(∀x P (x)) is same as ∃x (¬P (x))

¬(∃x P (x)) is same as ∀x (¬P (x))

Discrete Mathematics Lecture 2: Proof Techniques

Problems on Propositional Logic

The function ( (p ∨ (r ∨ q)

)∧ ¬

(p ∧ (¬q ∧ ¬r)

))is equal to which of the following functions:

A. q ∨ rB. ¬p ∨ (r ∧ q)C. (p ∨ q) ∨ rD. (p ∨ q) ∧ ¬(p ∨ r)E. (p ∧ r) ∨ (p ∧ q)

Discrete Mathematics Lecture 2: Proof Techniques

Negating a sentence

What is the negation of the sentence: “There is an university inUSA where every department has at least 20 faculty and atleast one noble laureate.”

A. There is an university in USA where every department hasless than 20 faculty and at least one noble laureate.

B. All universitis in USA where every department has at least20 faculty and at least one noble laureate.

C. For all universities in USA there is a department has lessthan 20 faculty or at most one noble laureate.

D. For all universities in USA there is a department has lessthan 20 faculty and at least one noble laureate.

Discrete Mathematics Lecture 2: Proof Techniques

Negating a sentence

What is the negation of the sentence: “There is an university inUSA where every department has at least 20 faculty and atleast one noble laureate.”

A. There is an university in USA where every department hasless than 20 faculty and at least one noble laureate.

B. All universitis in USA where every department has at least20 faculty and at least one noble laureate.

C. For all universities in USA there is a department has lessthan 20 faculty or at most one noble laureate.

D. For all universities in USA there is a department has lessthan 20 faculty and at least one noble laureate.

Discrete Mathematics Lecture 2: Proof Techniques

Propositional Logic and Predicate Logic

Every statement is either TRUE or FALSE

There are logical connectives ∨, ∧, ¬, =⇒ and ⇐⇒ .

Two logical statements can be equivalent if the twostatements answer exactly in the same way on every input.

To check whether two logical statements are equivalent onecan do one of the following:

Checking the Truthtable of each statementReducing one to the other using reductions

Discrete Mathematics Lecture 2: Proof Techniques

Propositional Logic and Predicate Logic

There are two important symbols: ∀ and ∃.Some statements can be defined using a variable.

For example: Px = “4x2 + 3 is divisible by 5”

We can have statements like: ∀x ∈ Z, 4x2 + 3 is divisible by5.

Or ∃x ∈ Z, 4x2 + 3 is divisible by 5.

Discrete Mathematics Lecture 2: Proof Techniques

Using Propositional Logic for designing proofs

A mathematical statement comprises of a premise (orassumptions). And when the assumptions are satisfied thestatement deduces something.

If A is the set of assumptions and B is the deduction thena mathematical statement is of the form

A =⇒ B

Now how to check if the statement if correct? And if it isindeed correct how to prove the statement?

Depending on whether A or B (or both) can be split intosmaller statements and how the smaller statements areconnected we can design different techniques for provingthe overall statement of A =⇒ B.

If indeed we can proof that the statement is correct thenwe can call it a Theorem.

Discrete Mathematics Lecture 2: Proof Techniques

Using Propositional Logic for designing proofs

A mathematical statement comprises of a premise (orassumptions). And when the assumptions are satisfied thestatement deduces something.

If A is the set of assumptions and B is the deduction thena mathematical statement is of the form

A =⇒ B

Now how to check if the statement if correct? And if it isindeed correct how to prove the statement?

Depending on whether A or B (or both) can be split intosmaller statements and how the smaller statements areconnected we can design different techniques for provingthe overall statement of A =⇒ B.

If indeed we can proof that the statement is correct thenwe can call it a Theorem.

Discrete Mathematics Lecture 2: Proof Techniques

Using Propositional Logic for designing proofs

A mathematical statement comprises of a premise (orassumptions). And when the assumptions are satisfied thestatement deduces something.

If A is the set of assumptions and B is the deduction thena mathematical statement is of the form

A =⇒ B

Now how to check if the statement if correct? And if it isindeed correct how to prove the statement?

Depending on whether A or B (or both) can be split intosmaller statements and how the smaller statements areconnected we can design different techniques for provingthe overall statement of A =⇒ B.

If indeed we can proof that the statement is correct thenwe can call it a Theorem.

Discrete Mathematics Lecture 2: Proof Techniques

Using Propositional Logic for designing proofs

A mathematical statement comprises of a premise (orassumptions). And when the assumptions are satisfied thestatement deduces something.

If A is the set of assumptions and B is the deduction thena mathematical statement is of the form

A =⇒ B

Now how to check if the statement if correct? And if it isindeed correct how to prove the statement?

Depending on whether A or B (or both) can be split intosmaller statements and how the smaller statements areconnected we can design different techniques for provingthe overall statement of A =⇒ B.

If indeed we can proof that the statement is correct thenwe can call it a Theorem.

Discrete Mathematics Lecture 2: Proof Techniques

Using Propositional Logic for designing proofs

A mathematical statement comprises of a premise (orassumptions). And when the assumptions are satisfied thestatement deduces something.

If A is the set of assumptions and B is the deduction thena mathematical statement is of the form

A =⇒ B

Now how to check if the statement if correct? And if it isindeed correct how to prove the statement?

Depending on whether A or B (or both) can be split intosmaller statements and how the smaller statements areconnected we can design different techniques for provingthe overall statement of A =⇒ B.

If indeed we can proof that the statement is correct thenwe can call it a Theorem.

Discrete Mathematics Lecture 2: Proof Techniques

Proof Techniques

To prove statement B from A.

Constructive Proofs

Proof by Contradiction

Proof by Contrapositive

Induction

Counter example

Existential Proof

Discrete Mathematics Lecture 2: Proof Techniques

Constructive Proof

To prove B from A.

[Direct Proof]: You directly proof A =⇒ B.

[Case Studies]: If A = X ∪ Y then A =⇒ B is same as(X =⇒ B) ∧ (Y =⇒ B).

Discrete Mathematics Lecture 2: Proof Techniques

Constructive Proof

To prove B from A.

[Direct Proof]: You directly proof A =⇒ B.

[Case Studies]: If A = X ∪ Y then A =⇒ B is same as(X =⇒ B) ∧ (Y =⇒ B).

Discrete Mathematics Lecture 2: Proof Techniques

Constructive Proof

To prove B from A.

[Direct Proof]: You directly proof A =⇒ B.

[Case Studies]: If A = X ∪ Y then A =⇒ B is same as(X =⇒ B) ∧ (Y =⇒ B).

Discrete Mathematics Lecture 2: Proof Techniques

Proof by Contradiction and Contrapositive

To prove B from A.

[Proof by Contradiction]: A =⇒ B is same as(¬B ∧A) = FALSE. That is, (¬B ∧A) gives acontradiction.

[Contraposetive]: A =⇒ B is same as (¬B =⇒ ¬A).

Discrete Mathematics Lecture 2: Proof Techniques

Proof by Contradiction and Contrapositive

To prove B from A.

[Proof by Contradiction]: A =⇒ B is same as(¬B ∧A) = FALSE. That is, (¬B ∧A) gives acontradiction.

[Contraposetive]: A =⇒ B is same as (¬B =⇒ ¬A).

Discrete Mathematics Lecture 2: Proof Techniques

Exercise on Contrapositive

What is the contrapositive of the following statement:

Given a finite family of convex sets C1, C2, ..., Cn in Rd (wheren ≥ d+ 1) such that if the intersection of every d+ 1 of thesesets is non-empty, then the whole collection has a non-emptyintersection.

Discrete Mathematics Lecture 2: Proof Techniques

Exercise on Contrapositive

What is the contrapositive of the following statement:

For every α (where 0 < α ≤ 1), there exists β = β(d, α) withthe following property. Let C1, C2, ..., Cn be convex sets in Rd

(where n ≥ d+ 1). If at least α(

nd+1

)of the collection of

subfamilies of size d+ 1 has a non-empty intersection, thenthere exists a point contained in at least βn sets.

Discrete Mathematics Lecture 2: Proof Techniques

Induction

If we have to prove “∀n ≥ r P (n) is True.”

Let us induct on n.

Base Case: Prove that the P (r) is true.

Induction Hypothesis: Let for some k ≥ r P (k) is true.

Inductive Step: We want to show, assuming IH, P (k + 1) istrue.

By induction we have proved that ∀n ≥ r P (n) is true.

Discrete Mathematics Lecture 2: Proof Techniques