By Oscar Levin

Note, this can be the corrected Fall 2015 version. a brand new version should be on hand August 2016

This light creation to discrete arithmetic is written for first and moment yr math majors, in particular those that intend to educate. The textual content all started as a suite of lecture notes for the discrete arithmetic direction on the collage of Northern Colorado. This path serves either as an creation to subject matters in discrete math and because the "introduction to evidence" path for math majors. The path is mostly taught with a large number of scholar inquiry, and this article is written to assist facilitate this.

Four major subject matters are coated: counting, sequences, common sense, and graph idea. alongside the way in which proofs are brought, together with proofs via contradiction, proofs through induction, and combinatorial proofs. The e-book includes 299 routines, all with ideas (or no less than a hint), in addition to forty five extra extra concerned difficulties appropriate for homework. There also are Investigate! difficulties in the course of the textual content to aid energetic, inquiry established learning.

• it really is written for use in an inquiry wealthy course.
• It is written for use in a direction for destiny math teachers.
• it really is open resource, with low-cost print variations and unfastened digital editions.

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Extra info for Discrete Mathematics: An Open Introduction

Sample text

What if f 1 a 2 a 3 and g b 13. Consider the function f : Z → Z given by f ( n ) (a) Is f injective? Prove your answer.   n + 1 n − 3  a 5 b 6 c ? 7 if is even if is odd. (b) Is f surjective? Prove your answer. 14. At the end of the semester a teacher assigns letter grades to each of her students. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bĳective, or neither? 15. In the game of Hearts, four players are each dealt 13 cards from a deck of 52.

There are elements in the codomain which are not in the range. For example, no n ∈ Z gets mapped to the number 1 (the rule would say that 31 would be sent to 1, but 31 is not in the domain). In fact, the range of the function is 3Z (the integer multiples of 3), which is not equal to Z. 2. g is not surjective. There is no x ∈ {1, 2, 3} (the domain) for which g ( x ) b, so b, which is in the codomain, is not in the range. Notice that there is an element from the codomain “missing” from the bottom row of the matrix.

The domain and codomain are both the set of integers. However, the range is only the set of integer multiples of 3. 2. g : {1, 2, 3} → { a, b, c } defined by g (1) c, g (2) a and g (3) a. The domain is the set {1, 2, 3}, the codomain is the set { a, b, c } and the range is the set { a, c }. Note that g (2) and g (3) are the same element of the codomain. This is okay since each element in the domain still has only one output. 3. h : {1, 2, 3} → {1, 2, 3} defined as follows: 1 2 3 1 2 3 This means that the function f sends 1 to 2, 2 to 1 and 3 to 3: just follow the arrows.