True or False

  • The empty set is a subset of every set TRUE, by definition!
  • The empty set is an element of every set FALSE, though it is true sometimes!
  • There is a surjective map from the set of the standard \(52\) playing cards to the set of U.S. states TRUE, for there to be a surjection the domain must have cardinality greater than or equal to the codomain.
  • There is a surjective map from the set of the standard \(52\) playing cards to the set of integers \(1,2,\dots,13\) TRUE, since the cardinality of the domain is larger than the cardinality of the codomain. One example is the function sending every card to its face value (A=1,J=11,Q=12,K=13)
  • We use axiomatic set theory to get around problems with paradoxes introduced by naive set theory, such as Russel’s paradox TRUE, as discussed in class and in Gallery of the Infinite

Example Section

For each of the following problems, give an example

  • A partition of 10
\[(3,3,3,1)\]
  • A set with exactly \(99\) elements Instead of writing one out, let’s use set builder notation!
\[\{x : x\in \mathbb N,\ 1\leq x\leq 99\}.\]
  • Sets \(A\) and \(B\) such that simultaneously \(B\subseteq A\) and \(B\in A\)
\[B = \{1\},\ \ A = \{1,2,3,\{1\}\}.\]
  • A function which is injective but not bijective
\[f: \{1,2,3\}\rightarrow \{1,2,3,4\},\ \ f(k) = k\]
  • A function which is surjective but not bijective
\[f: \mathbb R\rightarrow [0,\infty),\ \ f(x) = x^2\]
  • Five different sets, all of which are countably infinite
\[\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb {Z}^2,\ \{x: x\ \text{is an even integer}\}.\]
  • An uncountable set
\[\mathbb R\]
  • A set whose cardinality is larger than \(\mathbb R\)
\[\mathcal P(\mathbb R)\]

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 \(\{\{1,\heartsuit,\Delta\},2,\diamondsuit\}.\)

They have the same size!

  • The set of \(M\) of living or dead (biological) human mothers or the set \(C\) of living or dead (biological) human children

The set of children is larger than the set of mothers since each child has a mother but mothers can have multiple children.

  • The set \(\mathbb N\) of natural numbers of the set \(\mathbb Q\) of rational numbers

They set of rational numbers is countable, so they must have the same cardinality.

  • The set \((0,1)\) of real numbers between \(0\) and \(1\) or the set \(\mathbb R\) of all real numbers

They have the same cardinality, as we showed in class. Do you remember how?

  • The set \(\{\mathbb R,\mathbb Z\}\) or the set \(B\) of all even integers.

The first set only has two elements! Therefore the set \(B\) is larger.