box-o-sand/oldstuff/math-replay/algebra/new-definitions-1.haml

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    %title New Definitions 1
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    %h2 $x `not` $y = 8x - 3y$:
    :eqnarray
      x `not` y & = & 8x - 3y \\
      6 `not` -2 & = & 8(6) - 3(-2) \\
      6 `not` -2 & = & 48 + 6 \\
      6 `not` -2 & = & 54 \\

    %h2 $x `star` y = 8y + 2$:
    :eqnarray
      x `star` y & = & 8y + 2 \\
      -2 `star` 5 & = & 8(5) + 2 \\
      -2 `star` 5 & = & 40 + 2 \\
      -2 `star` 5 & = & 42 \\

    %h2 $x `diamond` y = xy + 3x - y$:
    :eqnarray
      x `diamond` y & = & xy + 3x - y \\
      3 `diamond` -2 & = & (3)(-2) + 3(3) - (-2) \\
      3 `diamond` -2 & = & -6 + 9 + 2 \\
      3 `diamond` -2 & = & -6 + 11 \\
      3 `diamond` -2 & = & 5 \\

    %h2 $x `not` y = x + 3y$:
    :eqnarray
      x `not` y & = & x + 3y \\
      -4 `not` 6 & = & -4 + 3(6) \\
      -4 `not` 6 & = & -4 + 18 \\
      -4 `not` 6 & = & 14 \\

    %h2 $x `donut` y = 3x - 3y$:
    :eqnarray
      x `donut` y & = & 3x - 3y \\
      0 `donut` 0 & = & 3(0) - 3(0) \\
      0 `donut` 0 & = & 0 - 0 \\
      0 `donut` 0 & = & 0 \\