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Chapter 1.1

# COSI 29a Chapter Notes - Chapter 1.1: Exclusive Or, Contraposition, Logical Biconditional

by OC1283201

School

Brandeis UniversityDepartment

Computer ScienceCourse Code

COSI 29aProfessor

Cherniack MitchChapter

1.1This

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Logic

Pages 1 – 20

Propositions

A declarative sentence that is either true or false.

The negation of a proposition (p) is denoted by ¬p (aka “not p”).

Ex: The negation of “Today is Friday” is “Today is not Friday”.

p q is called the conjunction of p and q.

p q (“p or q”) is called the disjunction of p and q.

The exclusive disjunction of p and q is denoted p ⨁ q (either p or q is true but not both).

Implications

The implication p → q (if p, then q) is only false when p is true and q is false (if the condition p

is met then q must be also met).

Ex: “If you get 100%, then you will get an A.” This statement is only false if you get 100% (p)

but you do not get an A (q).

The proposition q → p is the converse of p → q.

The proposition ¬p → ¬q is the inverse of p → q.

The proposition ¬q → ¬p is the contrapositive of p → q. The contrapositive has the same truth

value as p → q. Since they have the same truth value they are called equivalent.

The converse and inverse are equivalent.

The biconditional p ↔ q is true when both p and q have the same truth values. p ↔ q is

equivalent to (p → q) (q → p).

Precedence of Logical Operators

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