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Chapter 1: The Foundations: Logic and Proofs

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Chapter 1: The Foundations: Logic and Proofs

1.1 Propositional Logic

1.2 Propositional Equivalences

1.3 Predicates and Quantifiers

1.4 Nested Quantifiers

1.5 Rules of Inference

1.6 Introduction to Proofs

1.7 Proof Methods and Strategy

1.1: Propositional Logic

Propositions: A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.

Example 1:

All the following declarative sentences are propositions:

Washington D.C., is the capital of the USA.

2. Toronto is the capital of Canada

3. 1+1=2.

4. 2+2=3.

Example 2:

Consider the following sentences. Are they propositions?

1. What time is it?

2. Read this carefully.

3. x+1=2.

4. x+y=z

We use letters to denote propositional variables (or statement variables).

T: the value of a proposition is true.

F: the value of a proposition is false.

The area of logic that deals with propositions is called the propositional calculus or propositional logic.

Let p and q are propositions:

Definition 1: Negation (Not)

Symbol: ¬

Statement:“it is not the case that p”.

Example:

P: I am going to town

¬P:

It is not the case that I am going to town;

I am not going to town;

I ain’tgoin’.

Definition 2: Conjunction (And)

Symbol:

The conjunction pq is true when both p and q are true and is false otherwise.

Example:

P - ‘I am going to town’

Q - ‘It is going to rain’

PQ: ‘I am going to town and it is going to rain.’

Definition 3: Disjunction (Or)

Symbol:

The disjunction pq is false when both p and q are false and is true otherwise.

Example:

P - ‘I am going to town’

Q - ‘It is going to rain’

P Q: ‘I am going to town or it is going to rain.’

Definition 4: Exclusive OR

Symbol:

The exclusive or of p and q, denote pq, is true when exactly one of p and q is true and is false otherwise.

Example:

P - ‘I am going to town’

Q - ‘It is going to rain’

P Q: ‘Either I am going to town or it is going to rain.’

Definition 5: Implication

If…. Then….

Symbol:

The conditional statement pq is false when p is true and q is false, and true

P is called the hypothesis and q is called the conclusion.

Example:

P - ‘I am going to town’

Q - ‘It is going to rain’

P Q: ‘If I am going to town then it is going to rain.’

Equivalent Forms

If P, then Q

P implies Q

If P, Q

P only if Q

P is a sufficient condition for Q

Q if P

Q whenever P

Q is a necessary condition for P

Note: The implication is false only when P is true and Q is false!

‘If the moon is made of green cheese then I have more money than Bill Gates’ (?)

‘If the moon is made of green cheese then I’m on welfare’ (?)

‘If 1+1=3 then your grandma wears combat boots’ (?)

‘If I’m wealthy then the moon is not made of green

cheese.’ (?)

‘If I’m not wealthy then the moon is not made of green cheese.’ (?)

More terminology

QP is the CONVERSE of P Q

¬ Q ¬ P is the CONTRAPOSITIVE of P Q

¬ P ¬ Qis the inverse of P Q

Example:

Find the converse of the following statement:

R: ‘Raining tomorrow is a sufficient condition for my not going to town.’

Procedure

Step 1: Assign propositional variables to component propositions

P: It will rain tomorrow

Q: I will not go to town

Step 2: Symbolize the assertion

R: P Q

Step 3: Symbolize the converse

Q P

Step 4: Convert the symbols back into words

‘If I don’t go to town then it will rain tomorrow’

Homework: Find converse and contrapositive of statements above.

Definition 6: Biconditional

‘if and only if’, ‘iff’

Symbol:

The biconditional statement pq is true when p and q have the same truth value, and is false otherwise.

Biconditional statements are also called bi-implications.

Example:

P - ‘I am going to town’

Q - ‘It is going to rain’

P Q: ‘I am going to town if and only if it is going to rain.’

Translating English

Breaking assertions into component propositions - look for the logical operators!

Example:

‘If I go to Harry’s or go to the country I will not go

shopping.’

P: I go to Harry’s

Q: I go to the country

R: I will go shopping

If......P......or.....Q.....then....not.....R

(P Q) ¬ R

Constructing a truth table

one column for each propositional variable

one for the compound proposition

count in binary

n propositional variables = 2nrows

Construct the truth table for

(P ¬ Q) (PQ)

HW: Construct the truth table for (P Q) ¬ R

What is the real meaning of ¬ PQ ?

a) (¬ P) Q

b) ¬ (PQ)

What is the real meaning of PQR ?

a) (PQ)R

b) P(QR)

What is the real meaning of P QR ?

a) (P Q)R

b) P (QR)

Logic and Bit Operations

Example 20

Find the bitwise OR, bitwise AND, and bitwise XOR of the bit strings

01 1011 0110 and

11 0001 1101.

Logic Puzzles

Example 18:

There are two kind of inhabitants, knights, who always tell the truth, and their opposites, knaves, who always lie. You encounter two people A and B. What are A and B if A says “B is a knight” and B says “The two of us are opposite type”?

- Inverse
- Converse
- Contrapositive

Terms

Proposition

Negation

Conjection

Disjunction

Exclusive OR

Implication