Section 1.3 Quantifiers
Guiding Questions.
In this section, we’ll seek to answer the questions:
Many reasonable mathematical sentences do not meet our
definition of a statement. For instance,
\(x + 5 = 7\) is a declarative sentence, but its truth value depends on
\(x\text{;}\) when
\(x = 2\text{,}\) the statement is true, and when
\(x = 57\) it is not. We say that such a sentence is
open.
Definition 1.3.1.
An open sentence (or predicate) is a sentence \(P(x_1, x_2, \ldots, x_n)\) depending on variables \(x_1, x_2, \ldots, x_n\) with the property that the sentence becomes a statement when values are assigned to the variables, or a domain is specified for each variable.
That is to say, we can turn an open sentence into a statement in one of two ways. First, we could assign specific values to the variables. However, our typical approach will be to quantify the variables by assigning them to a particular domain, such as the real numbers or positive integers, and making one of two assertions: that the quantified statement is true for every object in the domain, or for some object in the domain.
Definition 1.3.2.
The existential quantifier, denoted with the symbol \(\exists\text{,}\) is the phrase “there exists” (or equivalent). The universal quantifier, denoted with the symbol \(\forall\text{,}\) is the phrase “for every” (or equivalent).
When writing a careful mathematical statement, it is often helpful to use quantifiers and mathematical symbols to clearly and concisely communicate an idea, such as
\begin{equation*}
(\forall x \in \R) (x^2 \ge 0),
\end{equation*}
which states that for every real number
\(x\text{,}\) the square of
\(x\) is nonnegative.
It can also be tempting to use the quantifier notation as an abbreviation when working through the scratchwork necessary to solve a problem. As a rule, however, one should
not use quantifier notation in a final write-up for a problem
unless the problem is about quantifiers.
Finding the right balance between mathematical symbols and ordinary writing takes time and practice. For more suggetions, see
Appendix A.
Activity 1.3.4.
Translate each quantified statement below to English. As best you can tell, is the statement true or false? Why?
\((\exists a\in\Z ) (2\cdot a = 2)\) (note that the symbols “\(a\in \Z\)” means that \(a\) is in the set \(\Z\) of integers, i.e., positive and negative whole numbers, and 0)
\(\displaystyle (\forall a\in \Z) (2\cdot a = 2)\)
\(\displaystyle (\forall x\in \Z, \exists y\in\Z)(x+y=0)\)
\(\displaystyle (\exists y\in \Z, \forall x\in \Z)(x+y=0)\)
Activity 1.3.6.
Translate the following from English to use mathematical notation in the style of
Activity 1.3.4. Choose variables as appropriate.
The square of every nonzero integer is greater than or equal to 1.
There exists a real number whose square is 2.
The cube of every positive integer \(m\) is greater than its square.
There is an integer \(x\) for which there is an integer \(y\) such that \(x\cdot y = -1\text{.}\)
Every real number is positive, negative, or zero.
We will often find ourselves needing to negate quantified statements. This should be done with care, especially as the statements get more complicated.
Problem 1.3.7.
Consider the statement “Every integer is positive.” One way to negate this statement is to say “It is not the case that every integer is positive,” but this style of negation is somewhat clumsy. Find a better way to negate the statement.
What does your answer to the previous question suggest about how to negate a universally quantified statement? Test your hypothesis on another univerally quantified statement of your own choosing.
Find an elegant negation of the statement “There is an integer \(x\) satisfying \(2x = 1\)” that uses a univeral quantifier.
Can the negation of any existentially quantified statement be written in terms of a universal quantifier?
Activity 1.3.8.
Negate the following statements. As best you can tell, is the statement true or false? Why?
\(\displaystyle (\exists a\in\Z ) (2\cdot a = 2)\)
The square of every nonzero integer is greater than or equal to 1.
\(\displaystyle (\forall a\in \Z) (2\cdot a = 2)\)
The cube of every positive integer \(m\) is greater than its square.
\(\displaystyle (\forall x\in \Z, \exists y\in\Z)(x+y=0)\)
There exists a real number whose square is 2.
There is an integer \(x\) for which there is an integer \(y\) such that \(x\cdot y = -1\text{.}\)
\(\displaystyle (\exists y\in \Z, \forall x\in \Z)(x+y=0)\)
Every real number is positive, negative, or zero.
