6.042J / 18.062J Mathematics for Computer Science (SMA by Nagpal, Radhika; Meyer, Albert R

By Nagpal, Radhika; Meyer, Albert R

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Example text

2. In every graph, there are an even number of vertices of odd degree. Proof. Partitioning the vertices into those of even degree and those of odd degree, we know � � � d(v) = d(v) + d(v) v∈V d(v) is odd d(v) is even The value of the lefthand side of this equation is even, and the second summand on the righthand side is even since it is entirely a sum of even values. So the first summand on the righthand side must also be even. But since it is entirely a sum of odd values, it must must contain an even number of terms.

A partition of a set A is a collection of subsets {A1 , . . , Ak } such that any two of them are disjoint (for any i �= j, Ai ∩ Aj = ∅) and such that their union is A. Let R be an equivalence relation on the set A. For an element a ∈ A, let [a] denote the set {b ∈ A given a ∼R b}. We call this set the equivalence class of a under R. We call a a representative of [a]. 3. The sets [a] for a ∈ A constitute a partition of A. That is, for every a, b ∈ A, either [a] = [b] or [a] ∩ [b] = ∅. Proof. Consider some arbitrary a, b ∈ A.

So (a − c) = (a − b) + (b − c) = (k1 + k2 )m. Therefore m | (a − c). The equivalence class of a is the set [a] = {b ∈ Z | a ≡ b (mod m)}, or {km + a | k ∈ Z}. It turns out that we can extend a lot of standard arithmetic to work modulo m. In fact, we can define notions of sum and product for the equivalence classes mod m. For example, we define [a] + [b], given two equivalence classes mod m, to be the equivalence class [a + b]. This is not as obvious as it seems: notice that the result is given in terms of a and b, two selected representatives from the equivalence classes, but that it is supposed to apply to the equivalence classes themselves.

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