CONVERSE, CONTRAPOSITIVE, INVERSEoThe proposition q → p is called the converse of p → q.o A conditional statement and its converse are NOT logically

equivalent.

oThe proposition ￢ p →￢ q is called the inverse of p → q.o A conditional statement and its inverse are NOT logically equivalent.

oIf a conditional statement is true, then its converse and inverse must also be true. This is not correct!

CONVERSE, CONTRAPOSITIVE, INVERSEoIf Howard can swim across the lake, then Howard can swim to the island.o Converse: o If Howard can swim to the island, then Howard can swim across the lake.o Inverse: o If Howard cannot swim across the lake, then Howard cannot swim to the

island.oIf today is Easter, then tomorrow is Monday.o Converse: If tomorrow is Monday, then today is Easter.o Inverse: If today is not Easter, then tomorrow is not Monday.

CONVERSE, CONTRAPOSITIVE, INVERSEoThe contrapositive of p → q is the proposition ￢ q →￢ p.o A conditional statement is logically equivalent to its

contrapositive!o If Howard can swim across the lake, then Howard can swim

to the island.o If today is Easter, then tomorrow is Monday.o If Howard cannot swim to the island, then Howard cannot swim

across the lake.o If tomorrow is not Monday, then today is not Easter.

CONVERSE, CONTRAPOSITIVE, INVERSE

Negations of If-Then Statements If my car is in the repair shop, then I cannot get to class. If Sara lives in Athens, then she lives in Greece.oNegation My car is in the repair shop and I can get to class. Sara lives in Athens and she does not live in Greece. oThe negation of an if-then statement does not start with the word if.

Simplifying Statement“you are hardworking and the sun shines, or you are hardworking and it rains.”p=“ you are hardworking “.q=“the sun shines”r= “it rains”

(p q) (p r) p (q r) using distributive law

“you are hardworking and the sun shines or it rains”

Translating English Sentences“If the moon is out and it is not snowing, then Sam goes out for a walk.”p=“Phyllis goes out for a walk”.q=“The moon is out”r= “It is snowing”

If the moon is out and it is not snowing, then Sam goes out for a walk.

“If it is snowing and the moon is not out, then Sam will not go out for a walk.”

prq )(

pqr )(

Translating English Sentences“You can access the Internet from campus only if you are a computer science major or you are not a freshman.”a=“You can access the Internet from campus“.c=“You are a computer science major”f= “You are a freshman”

a → (c ∨￢ f ).

Translating PropositionsoLet p, q, and r be the propositions:

p = “you have the flu”q = “you miss the final exam”r = “you pass the course”

oExpress the following propositions as an English sentence.op → q If you have flu, then you will miss the final exam.o~q → r If you don’t miss the final exam, you will pass the course.o~p → ~q → r If you neither have flu nor miss the final exam, then you

will pass the course.

BICONDITIONALSoIf it is hot outside you buy an ice cream cone, and if you buy an ice cream cone it is hot outside.

You buy an ice cream cone if and only if it is hot outside.o For you to win the contest it is necessary and sufficient that you have the only winning ticket.

You win the contest if and only if you hold the only winning ticket.oIf you read the news paper every day, you will be informed and conversely.

You will be informed if and only if you read the news paper every day.

oIt rains if it is a weekend day, and it is a weekend day if it rains.

It rains if and only if it is a weekend day.oThe train runs late on exactly those days when I take it. The train runs late if and only if it is a day I take the train.oThis number is divisible by 6 precisely when it is divisible by both 2 and 3.

This number is divisible by 6 if and only if it is divisible by both 2 and 3.

Predicateop=“is a student at Bedford College”oq =“is a student at.”• p & q are predicate symbols• p(x)=“x is a student at Bedford College.• q(x,y) =“x is a student at y.” x and y are predicate variables that take values in appropriate

sets.• When concrete values are substituted in place of

predicate variables, a statement results.oA predicate is a predicate symbol together with suitable predicate variables.

PredicateoA predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables.o The domain of a predicate variable is the set of all values that may be substituted in place of the variable.oAlso referred as propositional functions or open sentences

Predicate(Example)oPerson(x), which is true if x is a personoPerson(Socrates) = T oPerson(dolly-the-sheep) = FoLet U = Z, the integers = {. . . -2, -1, 0 , 1, 2, 3, . . .} P(x): x > 0 is the predicate. It has no truth value until the

variable x is bound.oExamples of propositions where x is assigned a value: P(-3) is false, P(0) is false, P(3) is true.

Truth Values of a PredicateoP(x)=“x2 > x”oDomain = set R of all real numbersoFind truth values of P(2), P( 1/2 ), and P(−1/2 )• P(2): 22 > 2, or 4 > 2. True• P(1/2):(1/2)2<1/2 or (1/4) <1/2 False• P(-1/2):(-1/2)2>-1/2 or (1/4) >-1/2 True

Truth Values of a PredicateoIf P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x. The truth set of P(x) is denoted

{x ∈ D | P(x)}“the set of all x in D such that P(x).”

Truth Values of a PredicateoLet R be the three-variable predicate R(x, y z): x + y= zoFind the truth value of R(2, -1, 5), R(3, 4, 7), R(x, 3, z)