Chapter 13 Cardinality of a set

1) Sets A and B have the same cardinality if... 2) The naturals and integers have the same/different cardinality because... 3) The naturals and reals do/don't have the same cardinality because...

1) if and only if there exists a bijection A→B 2) The same cardinalities because there exists a bijection between them. 3) Do not have the same cardinality because there exists no bijection between them.

What is the cardinality of ℚxℚ?

By theorems 13.4 and 13.5 ℚxℚ is countably infinite/ aleph naught.

What is the cardinality of an infinite subset of an countably infinite set?

Countably infinite (Theorem 13.8 pg. 230)

Prove that the cardinality of the powerset of a set, A, is bigger than the cardinality of A.

For finite sets, we know that the cardinality of the power set of A is 2 to the cardinality of A. For infinite sets, see pg. 229-230

Compare the cardinality of any set A to the cardinality of its powerset.

Given any set A (finite or infinite), the cardinality of A is less than the cardinality of its powerset (pg. 229)

If A and B are both countably infinite, then AUB ... prove it

Is countably infinite (Theorem 13.6 pg. 227)

What does the Cantor-Bernstein-Schroeder Theorem let us do?

It lets us show that two sets A and B have the same cardinality by finding injections f:A→B and g:B→A. This is useful because injections are often easier to find than bijections.

What does it mean for a an infinite set A to have a smaller cardinality than another infinite set B?

It means that there exists a bijection between A and B but no surjection (pg. 229)

If A and B are both countably infinite, then... prove it.

So is their Cartesian product, AxB. (Theorem 13.5). Just make a table with the elements of A on one side and the elements of B on the other. Where the columns and rows intersect write the ordered pair correspond to the elements of A and B. Then make a winding path through these ordered pairs.

Define countably infinite

Suppose A is a set. Then A is countably infinite if it has the same cardinality as the naturals, that is if there exists a bijection between the naturals and A.

Prove that an infinite subset of a countably infinite set is countably infinite.

Suppose A is an infinite subset of the countably infinite set B. Since B is countably infinite, we can write it's elements in an infinite list by theorem 13.3: b1, b2, b3,.... Then we can also write A's elements in list form by proceeding through the elements of B, in order, and removing the elements that aren't in A. Thus A can be written in list form, making it countably infinite.

If we know that some uncountable set U is a subset of another set A, what do we know about A?

That A is uncountable (Theorem 13.9 pg. 331)

Suppose B is an uncountable set and A is a set. Given that there is a surjective function f : A→B, what can be said about the cardinality of A?

That A is uncountable too.

Name sets that are countably infinite

The naturals, integers, rationals, etc.

If we can show that there exists an injection from (infinite sets) A to B, what can we say about the relative size of their cardinalities?

We can say that the cardinality of A is less than or equal to the cardinality of B (pg. 229)

Prove that If U⊆A, and U is uncountable, then A is uncountable.

We use contradiction and theorem 13.8 (pg. 231)

Prove that there are inifinte many primes

Write P as 2,3,5,7,11,13,... by Theorem 13.3, this is countable infinite.

Do the intervals (0,1) and (0,∞) have the same cardinality?

Yes, because there is a bijective function between these two sets

The set of rationals, ℚ, is/is not countably infinite. Prove it.

The set of rationals, ℚ, IS countably infinite (Theorem 13.4) (Pg. 224)

How do we define two sets as having the same cardinality?

There must be a bijective function between the two sets, f:A→B

Compare the cardinalities of the reals and the powerset of the naturals.

They are equal (Theorem 13.11 pg. 236)

Prove or disprove: if a set A is uncountable, then it's cardinality is equal to the cardinality of the reals.

This is false. Let A be the powerset of the reals. Then A is uncountable but it's cardinality is not equal to the cardinality of the reals.

True or false: the cardinality of the naturals is the same as the integers.

This is true because there exists a bijection between them.

Prove or disprove: If A⊆B and A is countably infinite and B is uncountable, then B−A is uncountable.

This statement is true. Suppose to the contrary that B-A is countably infinite. Then Au(B-A)=B is countably infinite (by theorem 13.6). But this a contradiction because B is uncountable (exercise 7 pg. 231)

Cantor-Bernstein-Schroeder Theorem

This tells us that given lAl≤lBl, and lBl≤lAl, then lAl=lBl (pg. 234). In other words if we can construct injections from A to B and from B to A then there is a bijection from A to B

A set is countably infinite if...

if and only if it's elements can be arranged in an infinite list (Theorem 13.3 pg. 223)

What does it mean for a set to be uncountable?

A set A is uncountable if it's cardinality is infinite but not countable infinite. That is, A is infinite and there exists no bijection from the naturals to A (pg. 223)

Compare the cardinalities of the naturals to the reals

Since there is no bijection between the naturals and the reals, their cardinality are not equal.

What is a surjection?

A surjective function (pg. 218)

What is a bijection?

A bijective function. I.e. a function that is both injective and surjective

What is the cardinality of the set of even integers E?

Aleph naught.

What is an injection?

An injective function (pg. 218)

How do we denote the cardinality of the naturals?

The cardinality of the natural numbers is ℵ0. That is |N| = ℵ0. This is pronounced "aleph naught"

Given n countably infinite sets A1,A2,A3,....,An with n≥2, their cartesian products have a cardinality of ... prove it.

aleph naught. In other words the cartesian product of all of them is countably infinite (Corollary 13.1 pg. 227)

The cardinality of the cross product of the integers and rationals is ....

countably infinite because the cardinality of both sets is countably infiinite, so their cross product is too by theorem 13.5

Show that equality of cardinalities is an equivalence relation on sets. In other words, show that equality of cardinalities is reflexive, symmetric, and transitive.

pg. 221