By Kevin Ferland

Discrete arithmetic combines a stability of idea and purposes with mathematical rigor and an obtainable writing sort. the writer makes use of more than a few examples to educate middle strategies, whereas corresponding routines enable scholars to use what they research. through the textual content, enticing anecdotes and themes of curiosity tell in addition to inspire rookies. The textual content is perfect for one- or two-semester classes and for college students who're usually arithmetic, arithmetic schooling, or machine technology majors. half I teaches pupil tips on how to write proofs; half II makes a speciality of computation and challenge fixing. the second one half the ebook can also be appropriate for introductory classes in combinatorics and graph concept.

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**Extra resources for Discrete Mathematics**

**Sample text**

In Exercises 21 through 28, determine whether the stated relations are true or false. 22. N ∈ Z. 21. 2 ∈ R. 23. {1} ∈ Z. 25. {2} ⊆ {1, 2, 3}. 27. ∅ ∈ {∅}. 24. 0 ⊆ Z. 26. 1 ∈ {{1}, {2}, {3}}. 28. ∅ ⊆ {∅}. In Exercises 29 through 34, list which of the relations ∈, ⊂, ⊆, = can replace the symbol . 29. {1} {1, 2}. 30. 2 {1, 2}. 31. {6, 7, 8} {8, 7, 6}. 32. {9} {9, {9, 10}}. 33. {3} {3, {3}, {{3}}}. 34. {4, 5} {{4}, {5}}. In Exercises 35 through 42, determine whether the given set is ﬁnite or inﬁnite.

E) 2 ∈ {{1}, {2}, 3} but {2} ∈ {{1}, {2}, 3} and {{2}} ⊆ {{1}, {2}, 3}. (f) 1 ∈ Z but 1 = Z and 1 Z. (g) 1 ∈ {1} but 1 = {1} and 1 {1}. 19 illuminate the difference between the number 1 and the set {1} containing just the number 1. In general, a set of the form {x} is called singleton x and should not be confused with the element x alone. Intervals are examples of subsets of R that are important enough to warrant their own notation. 9 Interval Notation Given real numbers a and b, we deﬁne the following intervals.

18. ∀ x ∈ R, ∃ y ∈ R such that x y ∈ Z. 19. ∀ m, n ∈ Z, m + n ∈ Z. 20. ∃ m, n ∈ Z such that 9m − 7n = 1. For Exercises 21 through 32, negate the given statements. 21. There is an integer whose reciprocal is also an integer. 22. Every integer power of 2 is a real number. 23. For every real number x, x 2 + 1 is positive. 24. There is an integer n such that 2n = 1024. 25. There is a natural number n such that, for every real number x, x n is nonnegative. 26. For any real number x, there is a real number y such that x + y = 0.