Statements with Multiple Quantifiers

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.

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

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?

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)

Example

"There is a real number with no reciprocal"

P(x , y ): y is the reciprocal of x

Is it true?

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)

Example

A student in the Discrete Math class owns a boatD: discrete math studentsB: collection of boatsQ(x , y): x owns y

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)

Example

The product of two negative real numbers is positive

T(x , y): x y >0

Negating Multiple QuantifiersAt each step consider one variable.

Other equivalences

Practice

Negate each of the following statements by following these steps: 1. Write them in symbolic logic2. Negate the symbolic expression3. Translate it back into English, as simple as

possible

• 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