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Chapter 1.1, 1.3

# COEN 19 Chapter Notes - Chapter 1.1, 1.3: Propositional Calculus, Logical Connective, Boolean Data Type

by OC2620276

School

Santa Clara UniversityDepartment

Computer EngineeringCourse Code

COEN 19Professor

LevinsonChapter

1.1, 1.3This

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Sect. 1.1

Propositions

A declarative sentence (declares a fact) that is either T/F, but not both.

e.g.

1. New York is capital of USA. | False proposition

2. 1 + 1 = 2 | True proposition

3. What time is it? | Not a proposition b/c not declarative

4. x + y = z | neither true or false

â€¢ propositional variables (sentential variables): letters or variables that represent

propositions. (p, q, r, s)

â€¢ Truth Value: T/F

â€¢ Atomic propositions: Propositions that cannot be expressed in terms of simpler

propositions.

â€¢ propositional calculus or propositional logic: area of logic that deals with propositions.

â€¢ Compound propositions: new propositions made by using logical operators to combine one

or more propositions.

â€¢ The notation for the negation operator is not standardized. Â¬p and p bar are

common notations to express the negation of p, other notations you might see are âˆ¼p, âˆ’p,

pâ€², Np, and !p.

â€¢ Connectives: logical operators used to form new propositions from two or more existing

propositions.

â€¢ The use of the connective or in a disjunction corresponds to an inclusive or. A disjunction is

true when at least one of the two propositions is true.

â€¢ The exclusive or of p and q is true when exactly one of p and q is true; it is false when both p

and q are true (and when both are false).

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If, Then Statements (Conditional Statements)

â€¢ The statement p â†’ q is a conditional statement because p â†’ q asserts that q is true on the

condition that p holds. A conditional statement is also an implication.

e.g. If I am elected president, then I will lower taxes. | This is only false if the elected candidate lies on

their campaign promise.

Converse, Contrapositives, & Inverse

â€¢ q â†’ p is the converse of p â†’ q.

â€¢ The contrapositive of p â†’ q is Â¬q â†’ Â¬p. | Note: only the contrapositive always has the same

truth value as p â†’ q

â€¢ The Â¬p â†’ Â¬q is the inverse of p â†’ q.

â€¢ When two compound propositions always have the same truth values, regardless of the

truth values of its propositional variables, we call them equivalent.

If & Only If (Biconditionals)

p â†” q is true when both p â†’ q and q â†’ p are true and is false otherwise.

Truth Tables of Compound Propositions

e.g. (p âˆ¨ !q) â†’ (p âˆ§ q)

.

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Precedence of Logical Operators

1. Not

2. And

3. Or

4. If, then

5. If & Only If

Logic & Bit Operations

Truth Value

Bit

T

1

F

0

â€¢ Boolean variable: its value is either T/F.

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