Give an example of a universal set U and a set A so that U is infinite and A is infinite and A^c (complement of A) is also infinite.
For this Big Problem, we’re getting really really big: infinity! Sets that are infinitely big
do interesting things. Here we’re going to experiment with some of them.
(1) Give an example of a universal set U and a set A so that U is infinite and A is infinite
and A^c (complement of A)
is also infinite.
(2) Give an example of a universal set U and a set A so that U is infinite and A is infinite
and A^c (complement of A)
is not infinite.
(3) Give an example of a set A and a set B so that A is infinite and B is infinite and
A ∩ B is also infinite.
(4) Give an example of a set A and a set B so that A is infinite and B is infinite and
A ∩ B is not infinite.
(5) If A is infinite and B is infinite, can you find an example where A ∪ B is not infinite?
Why or why not?
(6) If A is infinite and B is infinite, can you find an example where A−B is not infinite?
Why or why not?
(7) An Interesting Fact is that there are different sizes of infinity. For example, if A is an
infinite set, then P(A) is even bigger. Mathematicians argued about this for a long
time before agreeing it is true. How hard or easy is it for you to believe that there
are different sizes of infinity?
(8) What is your favorite infinite set?
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