2.4 Quantifiers

Objectives:

• Critically analyze proofs involving quantifiers

• Learn negations of mathematical statements involving quantifiers

• Apply knowledge of quantifiers to writing mathematical proofs

2.4.1 Predicates

As we saw in the last section, statements like “$x+2y=3$” appear regularly in mathematical writings. But statements like this, as written, are not propositions - they are neither true nor false but depend on the values of $x$ and $y$. Such statements are called predicates.

Example 2.4.1.

Which of the following statements are propositions, and which are predicates? Why?

(1) $x^2-4=0$

(2) For all $x\in ℝ$, $x^2-4=0$.

(3) .

(4) There exists an integer $b$ such that .

Notice how we were able to turn statements (1) and (3) from predicates to propositions (2) and (4) respectively by adding specific qualifications to what our variables denote. These “qualifications” - “for all” and “there exists” - are called quantifiers.

Definition 2.4.2.

• The universal quantifier ($\forall$, LaTeX $\backslash$forall) denotes “for all”. In logical notation, statement (2) reads $(\forall x\in ℝ)(x^2-4=0)$.

• The existential quantifier ($\exists$, LaTeX $\backslash$exists) denotes “there exists”. In logical notation, statement (4) reads .

Example 2.4.3.

Translate the following statement into logical notation: “for all positive values $\epsilon$, there exists a positive value $\delta$ such that, if $|x-c|$ is strictly between 0 and $\delta$, then $|f(x)-L|$ is less than $\epsilon$.” (This is the definition of the limit $L$ of a function $f(x)$ as $x$ approaches $c$.)

Blank space for you to write your work.

Example 2.4.4.

Determine whether the following statements are true, false, or neither. Prove each statement which is true, and disprove each statement which is false.

1. There exists a positive real number $x$ such that, if $x>0$, then $x^2>0$.

   Blank space for you to write your work.

2. For all real numbers $x\in ℝ$, if $x>0$, then, $x^2>0$.

   Blank space for you to write your work.

3. There exists an $n\in \mathbb{N}$ such that $n+1>3$.

   Blank space for you to write your work.

4. For all natural numbers $n\in \mathbb{N}$, $n+1>3$.

   Blank space for you to write your work.

5. There exists a real number $x$ such that $x>0$ and $x^2\le 0$.

   Blank space for you to write your work.

We recap what we have learned below:

1. To prove a "for all" statement, we must show a statement is true in general, not specifically for a certain value.
2. To disprove a "for all" statement, we must show a counterexample of our choosing, but we must verify that our counterexample indeed disproves the statement.
3. To prove a "there exists" statement, we must show an example of this object existing, then verify that this example works.
4. To disprove a "there exists" statement, we must show that "for all" relevant objects the statement does not hold true.

2.4.2 A “Contrapositive” for Quantifiers

Consider the following scenario:

A coach tells her players, “If everyone on the team runs a mile tonight, then I will buy the team pizza tomorrow.”

What would have to happen for the coach to be justified in not buying them pizza?

Blank space for you to write your work.

Now consider the following scenario:

A coach tells her players, “If one of you runs a mile tonight, then I will buy everyone pizza tomorrow.”

What would have to happen for the coach to be justified in not buying them pizza?

Blank space for you to write your work.

In each of these examples we are negating a “for all” or “there exists” statements. If $P(x)$ is the antecedent statement “player $x$ runs a mile” and $Q$ is the consequence statement “the coach buys the players pizza”, the first logical statement could be written as

$$(\forall x)P(x)\Rightarrow Q.$$

Well, what must be true if $\neg Q$ takes place? By contraposition, we know that $\neg (\forall x)P(x)$ is a true statement. But this is a bit awkward to say in English: “it is not the case that for all players $x$, player $x$ ran a mile”. We typically rephrase this in a different way: “there is some player $x$ who did not run a mile”. Logically this statement comes out to be $(\exists x)\neg P(x)$.

Similarly, the second statement is written as $((\exists x)P(x))\Rightarrow Q$. If $\neg Q$ is true, it must be the case that $\neg (\exists x)P(x)$ is also true! As we see in our answer, this can be written as “no one on the team runs a mile”. We can write this as $(\forall x)(\neg P(x))$.

This motivates the following theorem:

Theorem 2.4.5.

Let $P(x)$ be a predicate in some universe of discourse. Then

1. $\neg (\forall x)P(x)$ is logically equivalent to $(\exists x)(\neg P(x))$, and

2. $\neg (\exists x)P(x)$ is logically equivalent to $(\forall x)(\neg P(x))$.

Example 2.4.6.

Negate the following propositions, remembering not to use logical notation in your answers. These are called forming “useful denials” of these propositions. (Note: we are not proving or disproving these statements; we are just negating them!)

1. For all $n\in \mathbb{N}$, $n^2\ge 5$.

   Blank space for you to write your work.

2. For all $n\in \mathbb{Z}$, there exists $m\in \mathbb{Z}$ such that $n+m=0$.

   Blank space for you to write your work.

3. $(\forall x,y,z\in \mathbb{Z})((xy$ is even $\wedge yz$ is even$)\Rightarrow xz$ is even$)$

   Blank space for you to write your work.

2.4.3 Statements with Multiple Predicates

In the next sections we will see plenty of statements that involve multiple quantifiers. Before we get into these, let us experiment with a few examples.

Example 2.4.7.

Determine whether the following statements are true, false, or neither. Prove each statement which is true, and provide a counterexample for each statement which is false.

1. $(\forall x\in \mathbb{Z})(\forall y\in \mathbb{Z})x+y=0$

   Blank space for you to write your work.

2. $(\exists x\in \mathbb{Z})(\exists y\in \mathbb{Z})x+y=0$

   Blank space for you to write your work.

3. $(\exists x\in \mathbb{Z})(\forall y\in \mathbb{Z})x+y=0$

   Blank space for you to write your work.

4. $(\forall x\in \mathbb{Z})(\exists y\in \mathbb{Z})x+y=0$

   Blank space for you to write your work.

2.4.4 Proving with Quantifiers

In formal proofs you will not see mathematicians use abbreviations like $\forall$ and $\exists$, if for no other reason that it makes the proof harder to read. For the very same reason, you will see us label some parts of our proofs as "scratch work". These are considered parts of the problem to help us understand the statement, and they are typically never included in proofs.

Example 2.4.8.

Disprove the following statement. Remember, do not use logical notation in your proof.

$$(\forall x\in \mathbb{N})(\exists y\in \mathbb{N})(x-2y=0).$$

Blank space for you to write your work.

Example 2.4.9.

Consider the statement

$$(\forall y\in \mathbb{N})(\exists x\in \mathbb{N})(x-2y=0).$$

First, prove this statement. Then consider: if we switch the order of quantifiers, is the statement still true? Prove or disprove.

Blank space for you to write your work.