1. Statements with Multiple Quantifiers

2. When a statement contains more than one quantifier, we imagine the actions suggested by the quantifiers as being performed in the order in which the quantifiers occur. Also the variables may take values in different sets.

3. Possible Scenarios with Two variables(1) Let P(x, y) be a predicate with two variables, x in set D and y in set E ∀ x in set D, ∃ y in set E such that x and y satisfy property P(x, y). For each (or for all) x in D, there is one y in E that works

4. Example Definition: The reciprocal of a real number m is a real number n such that m*n = 1 "Every nonzero real number has a reciprocal" P(x, y): y is the reciprocal of x Is it true?

5. Let P(x, y) be a predicate with two variables, x in set D and y in set E ∃ x in set D, ∀ y in set E such that x and y satisfy property P(x, y). There is one x in D that works for any y in E Possible Scenarios with Two variables(2)

6. Example "There is a real number with no reciprocal" P(x, y ): y is the reciprocal of x Is it true?

7. Let P(x, y) be a predicate with two variables, x in set D and y in set E ∃ x in set D, ∃ y in set E such that x and y satisfy property P(x, y). There is an x in D and a y in E that work Possible Scenarios with Two variables(3)

8. Example A student in the Discrete Math class owns a boat D: discrete math students B: collection of boats Q(x, y): x owns y

9. Let P(x, y) be a predicate with two variables, x in set D and y in set E ∀ x in set D, ∀ y in set E such that x and y satisfy property P(x, y). Any x in D and any y in E work. Possible Scenarios with Two variables(3)

10. Example The product of two negative real numbers is positive T(x, y): x y >0

11. Negating Multiple Quantifiers At each step consider one variable.

12. Other equivalences

13. Practice Negate each of the following statements by following these steps: 1.Write them in symbolic logic 2.Negate the symbolic expression 3.Translate it back into English, as simple as possible

14. Each student has an ID number Some human beings do not have any disease There is a virus which is not detected by any antivirus program The product of two non-zero numbers is not zero