3.1 Predicates and Quantified Statements I

How many counter examples do we need to prove that the universal statement is false

1

when is an existential statement false

False if Q(x) is false for all values of x in D True if Q(x) is true for at least one value of x

When is a universal statement false

False when there is at least one value of Q(x) is false True if each value is true of X in D

Universal conditional statement form

For all if P(x), then Q(x)

what is truth set

If P(x) is a predicate and x has a domain D, the truthset of P(x) is the set of all elements of D that make P(x) true when they are substituted for x

What does the notation P(x) <-> Q(x) mean

P(x) and Q(x) have identical truth sets

what is a value that Q(x) is called? It proves the universal statement false

counter example

What does the notation P(x) -> Q(x) mean

every element in the truth set of P(x) is in the truth set of Q(x)

Multiple ways for universal quantifiers

for all for any for each for arbitrary for none (rewrite the statement to have for all)

Pros of exhaustion

it's easy if D is a smaller set it's very precise

Cons of exhaustion

it's impossible if D has infinitely many possiblities

What is exhaustion

method used to prove that a true universal statement trying all possiblities

How do you write a formal statement informally

no notations

what are quantifiers

obtain statements from predicates -- refer to quantities

What is the subject and predicate in this sentence: Rick Grimes was a sheriff's deputy

s: rick grimes p: was a sheriff's deputy

what is a predicate

sentence that contains a finite number of variables and becomes a statement when specific values are substituted for variables

what is a statement of a predicate variable

set of all values that may be substituted in place of the variable

what is predicate calculus

symbolic analysis of predicates and quantified statements

Multiple ways translate there exists?

there exists there is at least one there are some for some

Existential statement

there exists x in the domain such that Q(x)

What are the two types of quantifiers

universal and existential

what is the denotation of P(x)

{xED | P(x)}