3 2 Predicates and Quanti ed Statements II We de ne two quanti ed statements as truth values no matter what predicates are substituted for the predicate symbols and no matter what sets are used for the domain if they have identical Negations of Quanti ed Statements Consider the statement All students take calculus What is the negation of this statement De nition The negation of a statement of the form is logically equivalent to a statement of the form x D Q x Consider the statement Someone in this class is allergic to chocolate What is the negation of this statement De nition The negation of a statement of the form is logically equivalent to a statement of the form x D such that Q x 1 Write negations for the following statements Example 1 1 x R x2 1 2 y Z such that y2 7 3 No mathematicians are interesting 4 Some athletes are over the age of 40 5 The number 1357 is divisible by some integer between 1 and 37 Negating Condition Statements Recall that De nition P Q P x Q x The negation of a universal conditional statement is of the form x if P x then Q x x such that P x Q x x such that P x Q x 2 Example 2 Write negations for the following universal conditional statements 1 x if x 1 then x2 1 2 All actors making more than 26 million dollars will adopt a child from another country Note We can relate and by noting that x D Q x Q x1 Q x2 Q xn where xi D Similarly we can relate and by noting that x D such that Q x Q x1 Q x2 Q xn where xi D Example 3 Rewrite the following in terms of or 1 binary digits x x3 x 2 a binary digit x such that x x x Note that we say that the statement x D if P x then Q x is and only if P x is false for all x in our domain D For example suppose I said that all Jersey Shore cast members with a master s degree will go on to become astronauts This is vacuously true since none of the cast members have a master s degree if 3 Write a formal and informal contrapositive converse and inverse for the following statement If a Hogwart s student is sorted into Hu epu then he will lead an unimportant and boring life Variants of Universal Conditional Statements De nition Consider the statement x D if P x then Q x The is given by x D if Q x then P x The is given by x D if Q x then P x The is given by x D if P x then Q x Example 4 Formal Version of Statement Contrapositive Converse Inverse 4 De nition x r x is a for s x means x if r x then s x x r x is a for s x means x r x s x means x if r x then s x x if s x then r x Example 5 Rewrite the following statements as quanti ed conditional statements without the words necessary su cient and only if 1 Bravery is a necessary condition for being a Gry ndor quidditch player 2 Being a Malfoy is a su cient condition for being a Slytherin Potions Master 3 Students wear yellow only if they are in Hu epu 5 Group Work Complete the following problems in groups of 2 or 3 Write the converse contrapositive and inverse of the following statements 1 If an integer is greater than 3 then its reciprocal is less than 3 2 All girls at North Shore HS have been personally victimized by Regina George 6