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CSE 2500 Intro. to Discrete Systems

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### 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

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.

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

Different notations have been in use.

We will use the common math notation:

~ not

v or

^ and

→ implies (if … then …)

↔ if and only if

for all

exists

See the reverse of the text’s front cover

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 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

~highest

^

v

→, ↔ lowest

To avoid confusion - use ‘(‘ and ‘)’:

P ^ Q v X

(P ^ Q) v X

Determining Truth Values

Atomic statement: given

Compound statement: via meaning of the connectives

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

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

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

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

A → B

A is called:

Hypothesis

(or Antecedent)

B is called

Conclusion

(or Consequent)

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.

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

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

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

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

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

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

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

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

Suppose A and B are logically equivalent.

Consider the proposition (A ↔ B)

What can we say about it?

It is a tautology!

Why?

A B A ↔ B

T T T

F F T

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

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

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

Then all equivalences can be proven by definition using truth tables

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

F F T F T T T

F T F T T F F

T F F T F F T

T T F T F F F

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

~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

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!

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

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

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-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 …

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 …

(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?

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

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

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

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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