Introduction to Quantifiers

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Misc

Geneseo Insomnia Film Festival

A festival of short films created by student teams over a 24 hour period. See https://www.geneseo.edu/insomnia or contact Joe Dolce (dolce@geneseo.edu) for details.

Extra credit for anyone in this class who works on a team that does a non-tutorial film with (accurate) math as a plot element.

Register for GIFF sooner rather than later, but in no case after March 16.

Questions?

Problem Set

See handout for details.

Quantifiers

Section 2.4.

Basic Quantifiers

Using the following symbols (along with ∀, ∃, ℝ, ℤ, ℕ, ∈, etc.)...

• S is a set of students.
• D is a set of student ID numbers.
• n(s,d) is the predicate “s has ID number d”

Write the following in symbolic quantifier notation:

• Every ID number is an integer. (∀ d ∈ D)( d ∈ ℤ )
• Some ID number is less than 0. (∃ d ∈ D)(d < 0 )

Multiple Quantifiers

Using the same sets and predicate as above, write the following in symbolic form:

• Every student has an ID number. (∀ s ∈ S) ((∃ d ∈ D) n(s,d))
• Every student has the same ID number. (∃ d ∈ D)(∀ s ∈ S) (n(s,d)). Note that this differs from the statement above: the one above allows all students to have the same ID, but this one requires it.
• Every student has a unique ID.
  • (∀ s ∈ S)(∃ d ∈ D) ( n(s,d) and (∀ t ∈ S-{s})(not n(t,d)) )
  • In English, this says that for every student s there is an ID that s has and every student other than s does not have that ID.

Truth/Falsehood

Determine whether the following are true or false:

• (∀ n ∈ ℤ) ((∃ m ∈ ℤ)( n/m ∈ ℤ)). True
• (∃ m ∈ ℤ) ((∀ n ∈ ℤ)( n/m ∈ ℤ)). True.

The first says that every integer has at least one integer it can be divided by in order to produce an integer quotient, which is true because 1 and n both do that. The second says there is some “magical” integer that produces an integer quotient when divided into every other integer, which is true because 1 and -1 are such “magic” integers.

Key Ideas

The basic meanings of the quantifier symbols.

Chaining quantifiers together to form complex statements.

Determining informally when a quantified statement is true and when it is false.

Next

Negations and other advanced aspects of quantified statements.

The “Negations of Quantified Statements” subsection of section 2.4.

Next Lecture