Discrete Mathematics: An Open Introduction by Oscar Levin

By Oscar Levin

Note, this is often the corrected Fall 2015 version. a brand new version should be to be had August 2016

This mild advent to discrete arithmetic is written for first and moment yr math majors, specially those that intend to coach. The textual content started as a suite of lecture notes for the discrete arithmetic path on the collage of Northern Colorado. This path serves either as an creation to themes in discrete math and because the "introduction to evidence" path for math majors. The direction is mostly taught with a large number of pupil inquiry, and this article is written to aid facilitate this.

Four major themes are lined: counting, sequences, good judgment, and graph conception. alongside the way in which proofs are brought, together with proofs via contradiction, proofs through induction, and combinatorial proofs. The publication comprises 299 routines, all with ideas (or not less than a hint), in addition to forty five extra extra concerned difficulties compatible for homework. There also are Investigate! difficulties through the textual content to help lively, inquiry dependent learning.

While there are various wonderful discrete math textbooks to be had, this article has the next merits:

  • it really is written for use in an inquiry wealthy course.
  • It is written for use in a direction for destiny math teachers.
  • it truly is open resource, with not pricey print variations and loose digital editions.

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Extra resources for Discrete Mathematics: An Open Introduction

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It means that x is an element of A or x is an element of B (or both). That is, x ∈A∪B ⇔ x ∈ A ∨ x ∈ B. x ∈A∩B ⇔ x ∈ A ∧ x ∈ B. x∈A ⇔ ¬( x ∈ A). Similarly, Also, which says x is an element of the complement of A if x is not an element of A. There is one more way to combine sets which will be useful for us: the Cartesian product, A × B . This sounds fancy but is nothing you haven’t seen before. 3. Sets 26 plane. This is the set of all ordered pairs of real numbers ( x, y ). We can do this for any pair of sets, not just the real numbers with themselves.

Sometimes we denote this universe by U. Given this context, we might wish to speak of all the elements which are not in a particular set. We say B is the complement of A, and write, B A when B contains every element not contained in A. So, if our universe is {1, 2, . . , 9, 10}, and A {2, 3, 5, 7}, then A {1, 4, 6, 8, 9, 10}. Of course we can perform more than one operation at a time. For example, consider A ∩ B. This is the set of all elements which are both elements of A and not elements of B.

If A {1, 2} and B {2, 3, 4}, what is | AB |? What is | A × B |? Attempt the above activity before proceeding Relationships Between Sets We have already said what it means for two sets to be equal: they have exactly the same elements. Thus, for example, {1, 2, 3} {2, 1, 3} . ) Also, {1, 2, 3} {1, 1 + 1, 1 + 1 + 1} { I, II, III } since these are all ways to write the set containing the first three positive integers (how we write them doesn’t matter, just what they are). What about the sets A {1, 2, 3} and B {1, 2, 3, 4}?

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