Question

Give an example of two uncountable sets A and B such that $A\cap B$ is

1. finite.
2. countably infinite.
3. uncountable.

Short answer

1. The set {0} contains only 1 element, the set {0} is finite and thus $A\cap B$is finite.
2. The set N of nonnegative integers is countably infinite.
3. The set $R^{-}$of negative real numbers is uncountable.

Step-by-step solution

Step 1

1. FINITE DEFINITION:

A set is finite if it contains a limited number of elements (thus it is possible to list every single element in the set).

For example, the set $A=R^{+}\cup \left\{0\right\}$(nonnegative real numbers) is uncountable and the set $B=R^{-}\cup \left\{0\right\}$(non positive real numbers) is also uncountable.

Using the definition of the intersection and the fact that is the only element in both sets:

$A\cap B=\left(R^{+}\cup \left\{0\right\}\right)\cap \left(R^{-}\cup \left\{0\right\}\right)=\left\{0\right\}$

Since the set contains only 1 element, the set is finite and thus $A\cap B$is finite as well.

Step 2

1. COUNTABLY INFINITEDEFINITION::

A set is countably infinite if the set contains an unlimited number of elements and if there is a one-to-one correspondence with the positive integers.

For example, the set $A=(0,1)\cup N$(all real numbers between 0 and 1 and all nonnegative integers) is uncountable and the set $B=(2,3)\cup N$ (all real numbers between 2 and 3 and all nonnegative integers) is also uncountable.

Using the definition of the intersection and the fact that the non negative integers are the only elements in both sets:

$A\cap B=\left(\right(0,1)\cup N)\cap \left(\right(2,3)\cup N)=N$

The set N of nonnegative integers is countably infinite.

Step 3

1. UNCOUNTABLEDEFINITION:

A set is uncountable if the set is not finite or countably infinite.Intersection $A\cap B$: All elements that are both in A AND B in.

For example, the set A = R (real numbers) is uncountable and the set B =$R^{-}$ (negative real numbers) is also uncountable.

Using the definition of the intersection and the fact that the negative real numbers are the only elements in both sets:

$A\cap B=R\cap R^{-}=R^{-}$

The set $R^{-}$of negative real numbers is uncountable.