Appendix B Hints and Answers to Selected Odd Exercises

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Follow the proof in <xref ref="example-integers-binomial-theorem"/>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    The base case, <m>S(0): (1 + x)^0 - 1 = 0 \geq 0 = 0 \cdot x</m> is true.
    Assume <m>S(k): (1 + x)^k -1 \geq kx</m> is true.
    Then
    <md>

      <mrow>(1 + x)^{k + 1} - 1 &amp; = (1 + x)(1 + x)^k -1</mrow>

      <mrow>&amp; = (1 + x)^k + x(1 + x)^k - 1</mrow>

      <mrow>&amp; \geq kx + x(1 + x)^k</mrow>

      <mrow>&amp; \geq kx + x</mrow>

      <mrow>&amp; = (k + 1)x,</mrow>

    </md>

    so <m>S(k + 1)</m> is true.
    Therefore, <m>S(n)</m> is true for all positive integers <m>n</m>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    There is a nonabelian group containing six elements.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    The are five different groups of order 8.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>

    <md>

      <mrow>(aba^{-1})^n &amp; = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})</mrow>

      <mrow>&amp; = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}</mrow>

      <mrow>&amp; = ab^na^{-1}.</mrow>

    </md>

  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Since <m>abab = (ab)^2 = e = a^2 b^2 = aabb</m>, we know that <m>ba = ab</m>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>

    <m>H_1 = \{ id \}</m>, <m>H_2 = \{ id, \rho_1, \rho_2 \}</m>, <m>H_3 = \{ id, \mu_1 \}</m>, <m>H_4 = \{ id, \mu_2 \}</m>, <m>H_5 = \{ id, \mu_3 \}</m>, <m>S_3</m>.

  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    The identity of <m>G</m> is <m>1 = 1 + 0 \sqrt{2}</m>.
    Since <m>(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}</m>, <m>G</m> is closed under multiplication.
    Finally,  <m>(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)</m>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Since <m>a^4b = ba</m>, it must be the case that <m>b = a^6 b = a^2 b a</m>, and we can conclude that <m> ab = a^3 b a = ba</m>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>

    <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>, has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws"/>.  [Cross-reference is just a demo, content is not relevant.]  What happens if we relax the defintion and remove the parameter <m>n</m>?

  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    What did you see last time you went driving?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Maybe go out for a drive?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Do you know the acronym<ellipsis/><acro>ROY G BIV</acro> for the colors of a rainbow, and their order?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    What did you see last time you went driving?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Maybe go out for a drive?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    You can take a derivative on any one of the choices to see if it is correct or not, rather than using techniques of integration to find <em>a single</em> correct answer.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Dorothy will not be much help with this proof.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    For openers, a basis for a subspace must be a <em>subset</em> of the subspace.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Python boolean variables begin with capital latters.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>

    <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>, has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws"/>.  [Cross-reference is just a demo, content is not relevant.]  What happens if we relax the defintion and remove the parameter <m>n</m>?

  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    What did you see last time you went driving?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Maybe go out for a drive?
  </p>

</hint>

                            <hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

                              <p>
What did you see last time you went driving?
                              </p>

                            </hint>

                            <hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

                              <p>
Maybe go out for a drive?
                              </p>

                            </hint>