Break up into groups of three and work on these

  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 $U = \{\ a,\ \{a\},\ \{\ \}\ \}$. Write out all the subsets of $U$. Hint: there are eight subsets. Why? $|U|=3$ and for each of the three elements you have two choices: put the element in the subset, or leave it out. This gives $2\cdot 2\cdot 2 = 8$ possible outcomes.
  5. Let $W$ be a set. Is it possible for some $X$ to be both an element of $W$ and a subset of $W$? I.e. is it possible for $X\in W$ and $X \subseteq W$ to both be true?
  6. Let $A = \{1,2,3\}$. Write out the elements of $A\times A$.
  7. Let $A = \{1,2,3\}$. Write out the elements of $\{(u,v) \in A \times A\ \big|\ u + v = 4\ \}$.
  8. Write out the elements of $\{x \in \mathbb{R}\ \big|\ x^2 + x - 1 = 0 \}$
  9. Write out the elements of $\{ x \in \{0,1,\ldots,40\}\ \big|\ \exists y \in \mathbb{N} [\ y^2 = x\ ] \}$
  10. If $\Sigma$ is an alphabet, we denote the set of all strings over $\Sigma$ as $\Sigma^*$. Given alphabet $\Sigma$, give a short english description of the set $\{w\in\Sigma^*\ \big|\ w = w^R\}$.
  11. 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. Note: A SS& like 123-45-6789 should be represented as $(123,45,6789)$.
  12. Let $A$ be a finte subset of $\mathbb{R}$. Give an expression for the average of the elements of $A$.
  13. Let $D_i$, where $i \in \mathbb{N}$ be defined as: $ D_i = \{x \in \mathbb{N}\ \big|\ i = xy \mbox{ for some $y \in \mathbb{N}$} \} $ Give a (short!) english language description of the set: $P = \{i \in \mathbb{N}\ \big|\ |D_i| = 2\}$
    (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 $P$.