MATH 251 - 1.1 Propositions and Connectives

A ______ is a declarative statement that is either true or false.

Proposition

Is the following a proposition? Ex: Sit down

No

Is the following a proposition? Ex: The moon is made of cheese

Yes

A ___________ is a comprised of propositions and one or more of the following connectives

Compound proposition

What connective is this?

¬

Negation “not”

What connective is this?

∧

Conjunction “and”

What connective is this?

∨

Disjunction “or”

What connective is this?

→

Implication “if, then”

What connective is this?

↔

Biconditional “if and only if”

How would the following example change if P becomes ¬P?

Ex: My dog is the cutest dog.

My dog is NOT the cutest dog.

How would the following example change if P becomes ¬P?

Ex: The door is not open.

The door is open

2ⁿ rows where n = _________

Number of propositions

For a conjunction to be TRUE ….

Both propositions must be true

For a disjunction to be TRUE …

Either proposition must be true

Is P ∨ Q an inclusive or exclusive “or”?

Inclusive

Is P ⊕ Q an inclusive or exclusive “or”?

Exclusive

How is inclusive “or” P ∨ Q different from exclusive “or” P ⊕ Q?

Inclusive means one or both proposition are true and exclusive means only one proposition is true, not both.

A ____________ is a propositional form that is true for every assignment of truth values to its components.

Tautology

A _________ is a propositional form that is true for every assignment of truth values to its components.

Contradiction

We say two propositional forms are ___________ if they have the same truth tables.

Equivalent

What law is the following:

P and ~(~P)

Double Negation Law

What law is the following:

P ∨ Q and Q ∨ P

P ∧ Q and Q ∧ P

Commutative Laws

What law is the following:

P ∨ (Q ∨ R) and (P ∨ Q) ∨ R

P ∧ (Q ∧ R) and (P ∧ Q) ∧ R

Associative Laws

What law is the following:

P ∧ (Q ∨ R) and (P ∧ Q) ∨ (P ∧ R)

P ∨ (Q ∧ R) and (P ∨ Q) ∧ (P ∨ R)

Distributive Laws

What law is the following:

~(P ∧ Q) and ~P ∨ ~Q

~(P ∨ Q) and ~P ∧ ~Q

De Morgan’s Laws

A _______ of a proposition P is any proposition equivalent to ~P.

Denial