Below is some summary information on sets. You will be covering (or have covered) sets in Discrete Math, so this is just meant as a summary for folks that already have seen this material.

Sets

Problems on sets

  1. Let $A = \{x1, x3, x5, \ldots\}$. Give an element of $A$ other than $x1$, $x3$, and $x5$.
  2. Let $B = \{\{0\},\{1\},\{2\}\}$. True or false, the elements of $B$ are numbers.
  3. Let $C = \{\{\},\{a\},\{\{b\}\},\{c\}\}$. Explicitly list the elements of the set $\{x|\mbox{ there is a $y \in C$ such that $x \in y$}\}$.
  4. Let $F = \{0,5,10,15,20,25,\ldots, 50000\}$. Give a formal definition of $F$ without using ``$\ldots$''.
  5. Give a formal definition of the set of perfect squares.
  6. Let $D_i$ be the set of $i$-digit decimal natural numbers. Give a definition of the set of all social security numbers in terms of the $D_i$'s.
  7. Let $V$ be the set of vertices in a directed graph in which each edge is assigned a real number as a weight. Give a definition of the set of all possible edges in the graph. Remember: an edge is defined by the source and destination vertices, as well as the edge weight. Note: This is analogous to the definition of an \texttt{Edge} class in C++. What data elements would such a class contain?
  8. Let $A$ be a finte subset of $\mathbb{R}$. Give an expression for the average of the elements of $A$.
  9. Let $D_i$, where $i \in \mathbb{N}$ be defined as: $$ D_i = \{x \in \mathbb{N} | i = xy \mbox{ for some $y \in \mathbb{N}$} \} $$ Give a (short!) english language description of the set: $$P = \{i \in \mathbb{N}\ |\ |D_i| = 2\}$$ (Hint: Start by asking questions like this: Is 12 in $P$? Is 13 in $P$?) Note: This definition of the set $P$ is a lot like a computer program. I define a ``subroutine'' called ``$D_i$'' and use it in my ``main program'', which is the definition of $P$.