Section 3.1Beyond NumbersWhat Does Infinity Mean?

Understand simple things deeply.

Question of the Day

What comes to mind when you hear the word “infinity”?

Write your thoughts or questions on a piece of paper.

Provocative Questions

What is infinity?

What does it mean for a set to be finite?

How do you compare the size of two sets?

One-to-One Correspondence

Two collections of objects are equally numerous, precisely if there is a one-to-one correspondence between the elements of the two collections.

Section 3.2Comparing the Infinite

Pairing Up Collections via One-to-One Correspondence

Just because a specific attempt failed does not mean that the task at hand is impossible.

Question of the Day

How much bigger is the set of rational numbers than the set of natural numbers?

How many fractions are there between 0 and 1?

Cardinality

Cardinality – the “number” of things in a set.

Two sets have the same cardinality if there is a one-to-one correspondence between the elements of one set and the elements of the other set.

Bridge to Infinity

Natural Numbers: 1, 2, 3, 4, 5, 6, ….

Compare the cardinality of the natural numbers with the natural number 1 deleted.

Which set is larger?

Cardinality of other sets…

Compare the cardinality of the even natural numbers with the cardinality of the natural numbers.

Seeking higher cardinalities

Is there a set of numbers that has a greater cardinality than the cardinality of the natural numbers?

The Integers and Natural Numbers

Do the integers and the natural numbers have the same cardinalities?

The Integers and Natural Numbers

Rational Numbers

How many numbers are between two consecutive natural numbers?

Can you construct a one-to-one correspondence between the rationals and the natural numbers?

Rational Numbers

Section 3.3The Missing Member

Georg Cantor Answers: Some Infinities Larger Than Others?

Consider the counterintuituve.

Question of the Day

Is there an infinite set “bigger” than the Natural Numbers?

Cantor’s Theorem

There are more real numbers than natural numbers.

Section 3.4Travels Toward the Stratosphere of InfinitiesThe Power Set and the Question of an Infinite Galaxy of

Infinities

Make guesses, even if they are incorrect.

Question of the Day

Is there an infinite set “bigger” than the Real Numbers?

Sets

A set is a collection of objects.

The empty set is that contains nothing.

A subset is a set within the set.

How Many Subsets?

The subset count:

A set containing n elements has 2n subsets.

The Power Set

• The Power Set is the set that contains all subset of the set.

Cantor’s Power Set Theorem

Let S be a set (finite or infinite). Then the

cardinality of the power set of S, P(S), is strictly greater than the cardinality of S.

The Continuum Hypothesis

There is no cardinality between the cardinality of the set of natural numbers and the cardinality of the set of real numbers (sometimes referred to as the continuum).

Section 3.5Straightening Up the Circle

Exploring the Infinite Within Geometrical Objects

Examine arguments critically –

don’t just accept them.

Question of the Day

Are there more points in a square than on a line segment?

All Line Segments AreCreated Equal.

For any two line segments, the cardinality of points of one segment equals the cardinality of points of the other.

All Line Segments AreCreated Equal.

For any two line segments, the cardinality of points of one segment equals the cardinality of points of the other.

The Number Line is Like a Line Segment.

The cardinality of points between any two real numbers on a number line is equal to the cardinality of the set of all points on the real number line.

Squares Equal Lines

The cardinality of points inside a square is the same as the cardinality of points on a line segment.