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# Mathematics for Computer Science: MIT OCW 6.042

0.17MB. 0 audio & 1 images. Updated 2016-08-24.

## Description

## Sample (from 528 notes)

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Term |
Euler Tour |

Definition |
An \textit {Euler tour} is a defined to be a walk that traverses every edge in a graph exactly once and which starts and finishes at the same vertex. |

Subject |
DISCRETE MATH |

Tags |
chapter_5 euler_tour graph_theory paths walks |

Question |
What is the recurrence equation for the number of ways there are to climb $n$ stairs if you can either step up one stair or hop of two? |

Answer |
There is only 1 way to way to climb 0 stairs: do nothing. There is only 1 way to climb 1 stair: step up. In general, an ascent of $n$ stairs consists of either a step followed by an ascent of the remaining $n - 1$ stairs or a hop followed by an ascent of $n - 2$ stairs. So the total number of ways to climb $n$ stairs is equal to the number of ways to climb $n - 1$ plus the number of ways to climb $n - 2$. These observations define a recurrence:\begin{align*}f(0) &= 1 \qquad & \\f(1) &= 1 \qquad & \\f(n) &= f(n - 1) + f(n - 2) \qquad & \text{(for }n \geq 2)\end{align*}$f(n)$ denotes the number of ways to climb $n$ stairs. |

Subject |
DISCRETE MATH |

Tags |
chapter_12 fibonacci_series recurrences recursion series sums |

Term |
Neighbors |

Definition |
In any graph, the set $N(S)$ of \textit{neighbors} of some set $S$ of vertices is the set of all vertices adjacent to some vertex in $S$. That is: \[ N(S) ::= \{ r\, |\, \{s, r\} {\text { is a edge for some }} s \in S \} \] |

Subject |
DISCRETE MATH |

Tags |
chapter_5 graph_theory |

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