True or False

  • The empty set is a subset of every set
  • The empty set is an element of every set
  • There is a surjective map from the set of the standard \(52\) playing cards to the set of U.S. states
  • There is a surjective map from the set of the standard \(52\) playing cards to the set of integers \(1,2,\dots,13\)
  • We use axiomatic set theory to get around problems with paradoxes introduced by naive set theory, such as Russel’s paradox

Example Section

For each of the following problems, give an example

  • A partition of 10
  • A set with exactly \(99\) elements
  • Sets \(A\) and \(B\) such that simultaneously \(B\subseteq A\) and \(B\in A\)
  • A function which is injective but not bijective
  • A function which is surjective but not bijective
  • Five different sets, all of which are countably infinite
  • An uncountable set
  • A set whose cardinality is larger than \(\mathbb R\)

Comparing Sizes

For each of the following, determine which set is bigger. If they are the same size, then write that they are equal.

  • The set \(\{1,\heartsuit,\Delta\}\) or the set \(\{\{1,\heartsuit,\Delta\},2,\diamondsuit\}.\)
  • The set of \(M\) of living or dead (biological) human mothers or the set \(C\) of living or dead (biological) human children
  • The set \(\mathbb N\) of natural numbers of the set \(\mathbb Q\) of rational numbers
  • The set \((0,1)\) of real numbers between \(0\) and \(1\) or the set \(\mathbb R\) of all real numbers
  • The set \(\{\mathbb R,\mathbb Z\}\) or the set \(B\) of all even integers.