Math 3110 Spring 2010 -- Homework Problem Set #2

Due: Friday, February 5th
  1. Let \( f:X \to Y \) and \(g: Y \to X \). Suppose that \( f \circ g : Y \to Y \) is invertible.
    1. Show that \( f \) is surjective (i.e. onto).
    2. Show that \( g \) is injective (i.e. one-to-one).

  2. Consider the set of invertible \( 2 \times 2 \) matrices (this set is called \( \mbox{GL}_2(\mathbb{R}) \)).
    1. Is \( \mbox{GL}_2(\mathbb{R}) \) closed under matrix addition? If so, prove it. If not, give a counter-example.
    2. Is \( \mbox{GL}_2(\mathbb{R}) \) closed under matrix multiplication? If so, prove it. If not, give a counter-example.

  3. Consider the power set \( P = \mathcal{P}(S) \) where \( S = \{ 1,2,3,4 \} \). Let \( \approx \) be the relation defined by \( A\approx B \) iff \( |A|=|B| \) (that is the sizes of the sets \( A \) and \( B \) are equal).
    1. Show that \( \approx \) is an equivalence relation on \( P \).
    2. Find all of the equivalence classes of \( P \).
      [Hint: There are \( 2^{|S|}=16 \) subsets of \( S \). Equivalence classes should be sets of subsets of \( S \).]

  4. Let \( r \in \mathbb{R} - \{1\} \) and \( n \in \mathbb{Z}_{\geq 0} \). Prove the identity \[1 + r + r^2 + \cdots + r^n = \frac{1-r^{n+1}}{1-r} \] using induction.

  5. Let \(n \) be a non-negative integer. Use induction to show that \( 7^n-1 \) is an integer multiple of \( 6 \).