Cse 2500 intro to discrete systems
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CSE 2500 Intro. to Discrete Systems. Chun-Hsi (Vincent) Huang Dept. of Computer Science and Engineering University of Connecticut

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CSE 2500 Intro. to Discrete Systems

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Cse 2500 intro to discrete systems

CSE 2500Intro. to Discrete Systems

Chun-Hsi (Vincent) Huang

Dept. of Computer Science and Engineering

University of Connecticut

Notes are based on the book “Discrete Mathematics with Applications”, Third Edition, by Susanna Epp, in consultation with equivalent courses offered at other universities, including the Univ. of Alberta (V. Bulitko), the New York University (A. Bukharovich), as well as on-line materials provided by Brooks/Cole.


Logic

Logic

Mathematical logic is a tool for dealing with formal reasoning.

In a nutshell:

Logic does:

Assess if an argument is valid/invalid.

Logic does not directly:

Assess the truth of atomic statements.


Logic1

Logic

Logic can deduce that:

Storrs is in New England.

given these facts:

Storrs is in Connecticut.

Connecticut is a part of New England.

and the definitions of:

‘to be a part of’

‘to be in’

Logic does not know/concern whether these atomic statements actually hold true in real life!


Connectives

Connectives

Different notations have been in use.

We will use the common math notation:

~not

vor

^and

→implies (if … then …)

↔ if and only if

for all

exists

See the reverse of the text’s front cover


Logical statement

Logical Statement

(Definition, P6)

A statement form (or propositional form) is an expression made up of statement variables and logical connectives that becomes a statement when actual statements are substituted for the component statement variables. The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables.


Logical statement1

Logical Statement

A logical statement (or proposition) is either true or false, but not both.

Atomic:P, Q, X, Y, …

Negation: ~P, …

Conjunctive:P ^ Q, P ^ ~Q, …

Disjunctive:P v Q, P v (P ^ X),…

Conditional:P → Q

Biconditional:P ↔ Q


Order of precedence

Order of Precedence

~highest

^

v

→, ↔ lowest

To avoid confusion - use ‘(‘ and ‘)’:

P ^ Q v X

(P ^ Q) v X


Determining truth values

Determining Truth Values

Atomic statement: given

Compound statement:via meaning ofthe connectives

Suppose:P is trueQ is falseHow about:(P v Q)

Truth tables for P ^ Q, P v Q, ~P.


Conditionals

Conditionals

“If I go to Walmart tomorrow I will buy one pound of coffee there”

S = (I go to Walmart) →

(I buy one pound of coffee)

When is S true?

When I went to Walmart and bought one pound of coffee.

When I didn’t go there at all !

When is S false?

When I went there but didn’t buy one pound of coffee.


Truth table

Truth Table

N independent atomic variables

→2N rows (combinations)

N=20:220 ~ 1 million

Consider A → B:

A B A → B

T T T

T F F

F T T

F F T


More terminology

More Terminology

A → B

A is called:

Hypothesis

(or Antecedent)

B is called

Conclusion

(or Consequent)


Bi conditionals

Bi-Conditionals

“Michael will take CSE2500 if and only if Debra does so.”

S = (Michael takes CSE2500) ↔ (Debra takes CSE2500)

When is S true?

When Michael takes it and Debra takes it.

When Michael doesn’t take it and neither does Debra.

When is S false?

When one of them takes it but not the other.


Truth table1

Truth Table

A BA ↔ B

T T T

T F F

F T F

F F T


More on conditional statements

More on Conditional Statements

Suppose:

B = “I’ll buy a Ferrari.”

S = “The Ferrari is on sale.”

I will buy a Ferrari if it is on sale:

S → B

I will buy a Ferrari if and only if it is on sale:

S ↔ B

I will buy a Ferrari only if it is on sale:

~S → ~B

B → S


Sufficient necessary conditions

Sufficient, Necessary Conditions

Suppose X is a statement

A is a sufficient condition for X iff we can prove that:“A implies X”

A is necessary condition for X iff we can prove that: “X implies A”

A is a criterion for X iff we can prove that:“X holds if and only if A holds”


Contradictions tautologies

Contradictions & Tautologies

Contradiction:

A statement that is always false regardless of the values of its variables

Example: A ^ ~A

Tautology:

A statement that is always true regardless of the values of its variables

Example: A v ~A


Contingencies

Contingencies

What if I have a logical statement that is sometimes true and sometimes false?

It is called a contingency.

Example:

A ^ B

where A and B are independent variables.


Interpretation

Interpretation

In propositional logic, interpretation is an assignment of truth values of variables in the logical statement (expression).

Example:

Formula: A v B

Interpretation: A = true, B = false


Statement classifications

Statement Classifications

Tautology (T)

all interpretations satisfy the logical statement

Contradiction (C)

all interpretations falsify the logical statement

Contingency

some interpretations satisfy and some falsify the logical statement


Models counter models

Models/Counter-models

An interpretation is called a model for statement F iff:

It satisfies F (i.e., makes F=true)

An interpretation is called a counter-model for statement F iff:

It falsifies F (i.e., makes F=false)


Logical equivalence

Logical Equivalence

Statements A and B are logically equivalent when:

A holds if and only if B holds

Notation: A  B

Examples:

A v A is equivalent to A

A v ~A is equivalent to T (a tautology)


Equivalence tautology

Equivalence & Tautology

Suppose A and B are logically equivalent.

Consider the proposition (A ↔ B)

What can we say about it?

It is a tautology!

Why?

A BA ↔ B

T T T

F F T


More equivalences

More Equivalences

Alex is not unemployed.

Alex is employed.

P  ~(~P): Double-negation

It is not true that he is single and he is cute.

He is not single or he is not cute.

~(A ^ B)  ~A v ~B : De Morgan’s law

It is not true that he is single or he is cute.

He is not single and he is not cute.

~(A v B)  ~A ^ ~B : De Morgan’s law


Boolean algebra

Boolean Algebra

Page 14 presents Theorem 1.1.1 with 11 logical equivalences.

Example: Let’s prove that P v C  P (4)

P v C

 P v (P ^ ~P) by (5)

 (P v P) ^ (P v ~P) by (3)

 (P v P) ^ T by (5)

 P v P by (4)

 P by (7)


Proving equivalences

Proving Equivalences

Suppose P and Q can take on T and F only.

Then all equivalences can be proven by definition using truth tables

PQ ~(P v Q)~P ^ ~Q

FF T F T T T

FT F T T F F

T F F T F F T

T T F T F F F


Another example

Another Example

Prove P → Q  ~P v Q

Truth tables …


Uses of equivalences

Uses of Equivalences

Simplification

Suppose someone gives you

~P v (A→B) v ~(C v D→H) v P v X

and asks you to compute it for all possible input values.

You can either immediately draw a truth table with 27 = 128 rows

Or you can simplify it first.


Simplification

Simplification

~P v (A→B) v ~(C v D→H) v P v X

 ~P v P v (A→B) v ~(C v D→H) v X

 T v (A→B) v ~(C v D→H) v X

 T

The statement is a tautology.


Test drive

Test Drive

Let’s take our equivalence tool box for a spin …

What can we tell about

A → B

and its contraposition:

~B → ~A

?

They are equivalent!


Proof

Proof?

By truth-table

By logical equivalence

A → B

 ~A v B

 B v ~A

 ~B → ~A


Test drive1

Test Drive

Let’s try another one!

What can we tell about

A → B

and its inverse:

~A → ~B

?

They are not equivalent!

Counter-model: A is false and B is true…


Summary

Summary

Implication

A → B

is equivalent to:

~B → ~A : its contraposition

But is not equivalent to:

~A → ~B : its inverse

Or to:

B → A : its converse


Another lesson learned

Another Lesson Learned

Proving equivalences

via truth-tables

via other equivalences

Proving non-equivalence:

Finding an instantiation that makes one statement hold and the other doesn’t


Bi conditionals1

Bi-conditionals

How can we express:

A ↔ B

with ^, v, ~ ?

It is simply:

(A ^ B) v (~A ^ ~B)

How can we express it with →, Λ ?

Why, but of course it is just:

(A → B) ^ (B → A)


Another spin

Another Spin …

Ok, let’s try this fun ride:

(P → Q) ^ P

In English it would sound:

(If P is true then Q is true) and (P is true)

What does it tell us?

(One would intuitively think Q is true!)

Let’s prove/disprove it …


Proving

Proving …

(P → Q) ^ P

 (~P v Q) ^ P

 (~P ^ P) v (Q ^ P)

 F v (Q ^ P)

 Q ^ P

But we want just Q, not Q ^ P!

Is Q ^ P  Q?

No -- counter-model: Q=T and P=F …


What is the matter

What is the matter?

Not all arguments have the form of a chain of equivalences.

Example:

P→Q= “If Socrates is human then Socrates is mortal”

P=“Socrates is human”

A conjunction of these two is NOT equivalent to Q = “Socrates is mortal”

Why?

Name a dog Socrates. It is mortal (Q holds). But it is NOT human (P does not hold)…


Logical argument

Logical Argument

A collection of statements P1, …,Pn (premises) logically implies (entails) statement Q (conclusion)

if and only if:

Whenever all premises hold, the conclusion holds.

Example:

Premises:

P1= “If Socrates is human then Socrates is mortal”

P2 = “Socrates is human”

Conclusion:

Q = “Socrates is mortal”


Valid invalid arguments

Valid/Invalid Arguments

Suppose someone makes an argument:

P1, …,PN therefore Q.

The argument is called valid iff:

P1, …,PN logically imply Q.

That is:

Q must hold when all Pi hold.

Otherwise the argument is called invalid.


Relation to tautologies

Relation to Tautologies

We already know that statements A and B are equivalent iff (A ↔ B) is a tautology (i.e., holds for any interpretation)

How about entailment?

Statement A entails (logically implies) statement B iff (A → B) is a tautology.

In general: premises P1, …,PN entail (logically imply) Q iff (P1 ^ … ^ PN→ Q) is a tautology.

Refer to Table 1.3.1 on P40 of Textbook for a summary of Rules of Inference.


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