The Proof Page

by D. A. Kouba

Section 1.3- Universal Quantifier (For all x ...); Existential Quantifier (There exists x ...); Unique Existential Quantifier (There exists a unique x ...)

: A sentence containing one or more variables is called an .

: An open sentence is NOT a proposition because it is neither true nor false until all of its variables are replaced with values.

: The expression is an open sentence. Define to be the open sentence . Then means , which is a true proposition, and means , which is a false proposition.

: Define to be the open sentence . Then means , which is a true proposition, and means , which is a false proposition.

: The set of objects (or ) available for consideration (substitution) in an open sentence (or ) is called a . The set of elements (or ) which makes (or ) true is called the .

: Let the universe be the set of all real numbers for the open sentence . Then the truth set is . Let the universe be the set of all positive integers for the open sentence . Then the truth set is .

: Let be an open sentence with variable .

- 1.) The
is the sentence
(`` For all , ") and is true exactly when the truth set for is the entire universe. In other words,
*all*elements in the universe make true.

- 2.) The
is the sentence
(`` There exists such that ") and is true exactly when the truth set for is not empty. In other words, there exists
*at least one*element in the universe for which is true.

- 3.) The
is the sentence
(`` There exists a unique such that ") and is true exactly when the truth set for has exactly one element. In other words, there exists
*exactly one*element in the universe for which is true.

:

- 1.) Let the universe be the set of all real numbers and consider the open sentence . Consider the quantified sentence . This sentence is true since it is true for every element in the universe.
- 2.) Let the universe be the set of all complex numbers and consider the open sentence . Consider the quantified sentence . This sentence is false since it is false for some elements in the universe. For example, is false since .
- 3.) Let the universe be the set of all real numbers and consider the open sentence . Consider the quantified sentence . This sentence is true since it is true for at least one element in the universe. In fact, the truth set is .
- 4.) Let the universe be the set of all real numbers and consider the open sentences is an integer, and is a perfect square. Consider the quantified sentence . This sentence is true since it is true for at least one element in the universe. For example, and make the sentence true. The truth set contains infinitely many elements.
- 5.) (Pay close attention to the logic in this example.) Let the universe be the set of all real numbers and consider the open sentences is a natural number, and is a natural number. Consider the quantified sentence . This sentence is true since it is true for every element in the universe ! For example, if is a natural number, then is a true statement since both and are true. If is NOT a natural number, then is a still a true statement since is false ! Thus, is a true statement for every element in the universe.
- 6.) Let the universe be the set of all real numbers and consider the open sentence . Consider the quantified sentence . This sentence is true since it is true for exactly one element in the universe. It is true for .
- 7.) Let the universe be the set of all real numbers and consider the open sentence . Consider the quantified sentence . This sentence is true (since is an increasing function). If is any element in the universe, then is the unique solution to .

: Let the universe be the set of all animals. Consider the open sentences is a dog, chases cars, and eats vegetables. Rewrite each of the following common sentences in symbolic quantifier form. Find solutions HERE .

- 1.) Some dogs eat vegetables.
- 2.) All dogs chase cars.
- 3.) No dogs chase cars.
- 4.) There are some animals which chase cars but do not eat vegetables.

- 1.) Some cats like water.
- 2.) Some cats don't like water.
- 3.) No cats like water.
- 4.) All cats like water.

: Let be an open sentence with variable .

- a.) is equivalent to .
- b.) is equivalent to .

is TRUE

is FALSE

The truth set for is NOT the entire universe

There is at least one element in the truth set for which

is TRUE .

is FALSE

is TRUE

The truth set for is the entire universe

The truth set for is empty

is FALSE.

b.) The proof that is equivalent to is analogous to the proof of part a.).

: Let the universe be the set of all animals. Consider the open sentences is a dog , chases cars, and eats vegetables. Rewrite each of the following common sentences in symbolic quantifier form. Use Theorems 1.2 and 1.3 to write formal denials of these statements. Finish by writing a meaningful denial in ordinary English. Find solutions HERE .

- 1.) Some dogs eat vegetables.
- 2.) All dogs chase cars.
- 3.) No dogs chase cars.
- 4.) There are some animals, which chase cars but do not eat vegetables.

.

. . . . . . (By Theorem 1.3 a.)

. . . . . . (By Theorem 1.3 b.)

. . . . . . (By Theorem 1.2 d.)

. . . . . . (By Theorem 1.2 c.)

There exists an so that, for all , there exists a real number satisfying
but
.

- 1.) For all real numbers there is another real number so that . Consider the open sentence .
- 2.) For each real number there is another real number so that if , then or . Consider the open sentences , , and .

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Please e-mail your comments, questions, or suggestions to D. A. Kouba at
kouba@math.ucdavis.edu .