By Kenneth Bogart, Clifford Stein, Robert L. Drysdale
Discrete arithmetic for laptop technological know-how is the proper textual content to mix the fields of arithmetic and laptop technology. Written through prime teachers within the box of desktop technology, readers will achieve the talents had to write and comprehend the concept that of evidence. this article teaches the entire math, apart from linear algebra, that's had to reach machine technology. The publication explores the themes of uncomplicated combinatorics, quantity and graph idea, common sense and evidence ideas, and plenty of extra. applicable for giant or small type sizes or self learn for the prompted specialist reader. Assumes familiarity with information buildings. Early remedy of quantity thought and combinatorics permit readers to discover RSA encryption early and in addition to inspire them to exploit their wisdom of hashing and timber (from CS2) sooner than these issues are lined during this path.
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Additional resources for Final DRAFT 2004 --- Discrete Mathematics for Computer Science
Example text
The set of all lists of the four people. (a) How many lists are equivalent to a given one? (b) What are the lists equivalent to ABCD? (c) Is the relationship of equivalence an equivalence relation? (d) Use the Quotient Principle to compute the number of equivalence classes, and hence, the number of possible ways to seat the players. 4-3 We wish to count the number of ways to attach n distinct beads to the corners of a regular n-gon (or string them on a necklace). We say that two lists of the n beads are equivalent if each bead is adjacent to exactly the same beads in both lists.
Sm is a partition of S, the relationship that says any two elements x ∈ S and y ∈ S are equivalent if and only if they lie in the same set Si is an equivalence relation. The sets Si are called equivalence classes 4. Quotient principle. The quotient principle says that if we can partition a set of p objects up into q classes of size r, then q = p/r. Equivalently, if an equivalence relation on a set of size p has q equivalence classes of size r, then q = p/r. The quotient principle is frequently used for counting the number of equivalence classes of an equivalence relation.
3 below, we work out a special case. Suppose n = 5, k = 2. 6 says that 5 4 4 = + . 3. BINOMIAL COEFFICIENTS 21 Because the numbers are small, it is simple to verify this by using the formula for binomial coefficients, but let us instead consider subsets of a 5-element set. 7 says that the number of 2 element subsets of a 5 element set is equal to the number of 1 element subsets of a 4 element set plus the number of 2 element subsets of a 4 element set. But to apply the sum principle, we would need to say something stronger.