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Section 1.7 More with Quantifiers DeMorgan’s Law for Quantified Statements Announcements • Don’t forget to pay attention to the actual reading schedule. • Phones should be away unless we are using them in class. • You are supposed to be participating with your groups during the group activities. Activity #1 • Assume: – – – – the domain consists of integers O(x) is “x is odd” L(x) is “x < 10” G(x) is “x>9” Activity #1 • Given the prior assumptions, what is the truth value of the following statements. 1. 2. 3. 4. 5. x [ O(x) ] x [L(x) O(x) ] x [L(x) ¬ G(x) ] x [L(x) G(x)] x [L(x) G(x) Activity #1 • Assume: – the domain consists of integers – A(x) is “x<5” – B(x) is “x<7” Activity #1 • Given the prior assumptions, what is the truth value of the following statements. 1. 2. 3. 4. 5. x [ A(x) ] x [ A(x) B(x)] x [ A(x) B(x)] x [ A(x) B(x) ] x [ B(x) A(x) ] Activity #2 • How would you write the negation of: – All Americans eat cheeseburgers – There is a smart student at the University of Iowa. Negation of a Quantified Statement • You had to pay careful attention to negation with propositional logic (DeMorgan’s laws). – Predicate Logic has the same need for detail Negation of a Quantified Statement • Consider the following statement: – Every student in this course has taken a class in statistics. • This is a standard universal quantification: x D, Q(x) • What does this look like if I negate the statement: ¬ ( x D, Q(x) ) Negation of a Quantified Statement • Actually, they are called DeMorgan’s Laws for Quantifiers, and they LOOK like DeMorgan’s Laws for Propositional Logic ¬ ( x D, Q(x) ) x D, ¬ Q(x) ¬ ( x D, Q(x) ) x D, ¬ Q(x) Negation of a Universally Quantified Statement • Thus, the negation of a universal statement (“all are”) is logically equivalent to an existential statement with negation ¬ ( x D, Q(x) ) x D, ¬ Q(x) (“some are not” or “there is at least one that is not”) Negation of a Universally Quantified Statement • Think about this. If we say x D, Q(x) is false • What we are saying is: ¬ [ x D, Q(x) ] Negation of a Universally Quantified Statement ¬ [ x D, Q(x) ] ¬ [ Q(x1) Q(x2) Q(x3) … Q(xn) ] Equivalent expression/definition of the Universal Statement ¬ Q(x1) ¬ Q(x2) ¬ Q(x3) … ¬ Q(xn) DeMorgan’s Law of negating conjunction x D, ¬ Q(x) Equivalent expression of the Existential Statement Negation of an Existentially Quantified Statement • Thus, the negation of an existential statement (“some are”) is logically equivalent to a universal statement with negation ¬ ( x D, Q(x) ) x D, ¬ Q(x) (“none are” or “all are not”) Similarly x D, Q(x) is false ¬ [ x D, Q(x) ] ¬ [ Q(x1) Q(x2) Q(x3) … Q(xn) ] Equivalent expression/definition of the Existential Statement ¬ Q(x1) ¬ Q(x2) ¬ Q(x3) … ¬ Q(xn) DeMorgan’s Law of negating disjunction x D, ¬ Q(x) Equivalent expression of the Universal Statement Negation of a Quantified Statement • Thus, the negation of Every student in this course has taken a class in statistics. • Is not: No student in this class has taken a class in statistics • But There exists at least one student in this class who has not taken statistics Activity #2 • How would you write the negation of: – All Americans eat cheeseburgers – There is a smart student at the University of Iowa. Negations of Universal Conditional Statements • The negation of a universal conditional statement is: ¬ ( x, if P(x) then Q(x) ) x such that P(x) and ¬ Q(x) Activity #3 • real numbers x, if x2 >= 1 then x> 0 • For every student at UNI if they have been at UNI for at least two years then they are classified as a Junior. Vacuous Truth of Universal Statement • Every senior citizen in this room fought during world war II. • Is this a true statement or a false statement? Vacuous Truth of Universal Statement • To be FALSE then it’s negation must be TRUE – That’s the definition of negation. • But the negation of that statement is: – There exists at least one senior citizen in this room that did not fight in world war II. – This statement is clearly false (since there are no senior citizens in this room for which this is true) – This means that the original statement MUST be true