# SICP Exercise 2.11

The table for the min and max for each combination is as folows:

\(L_x\) | \(U_x\) | \(L_y\) | \(U_y\) | \(L_x L_y\) | \(L_x U_y\) | \(U_x L_y\) | \(U_x U_y\) | min | max |

+ | + | + | + | + | + | + | + | \(L_x L_y\) | \(U_x U_y\) |

+ | + | - | + | - | + | + | + | \(U_x L_y\) | \(U_x U_y\) |

+ | + | - | - | - | - | - | - | \(U_x L_y\) | \(L_x U_y\) |

- | + | + | + | - | - | + | + | \(L_x U_y\) | \(U_x U_y\) |

- | + | + | - | + | - | - | + | \(min(L_xU_y, U_x L_y)\) | \(max (L_x L_y, U_x U_y)\) |

- | + | - | - | + | + | - | - | \(U_x L_y\) | \(L_x L_y\) |

- | - | + | + | - | - | - | - | \(L_x U_y\) | \(U_x L_y\) |

- | - | - | + | + | - | + | - | \(L_x L_y\) | \(L_x U_y\) |

- | - | - | - | + | + | + | + | \(U_x U_y\) | \(L_x L_y\) |

So our procedure now becomes:

```
(define (mul-interval x y)
(define not-negative?
(lambda (x) (not (negative? x))))
(let ((lx (lower-bound x))
(ux (upper-bound x))
(ly (lower-bound y))
(uy (upper-bound y)))
(cond ((and (not-negative? lx)
(not-negative? ux)
(not-negative? ly)
(not-negative? uy)))
(make-interval (* lx ly)
(* ux uy)))
(cond ((and (not-negative? lx)
(not-negative? ux)
(negative? ly)
(not-negative? uy)))
(make-interval (* ux ly)
(* ux uy)))
(cond ((and (not-negative? lx)
(not-negative? ux)
(negative? ly)
(negative? uy)))
(make-interval (* ux ly)
(* lx uy)))
(cond ((and (negative? lx)
(not-negative? ux)
(not-negative? ly)
(not-negative? uy)))
(make-interval (* lx uy)
(* ux uy)))
(cond ((and (negative? lx)
(not-negative? ux)
(not-negative? ly)
(negative? uy)))
(make-interval (min (* lx uy) (* ux ly))
(max (* lx ly) (* ux uy)))
(cond ((and (negative? lx)
(not-negative? ux)
(negative? ly)
(negative? uy)))
(make-interval (* ux ly)
(* lx ly)))
(cond ((and (negative? lx)
(negative? ux)
(not-negative? ly)
(not-negative? uy)))
(make-interval (* lx uy)
(* ux ly)))
(cond ((and (negative? lx)
(negative? ux)
(negative? ly)
(not-negative? uy)))
(make-interval (* lx ly)
(* lx uy)))
(cond ((and (negative? lx)
(negative? ux)
(negative? ly)
(negative? uy)))
(make-interval (* ux uy)
(* lx ly))))))
```