By Harris Kwong

It is a textual content that covers the normal issues in a sophomore-level direction in discrete arithmetic: common sense, units, facts innovations, uncomplicated quantity thought, features, kinfolk, and basic combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in arithmetic, and courses the scholars via an in depth dialogue on how an explanation is revised from its draft to a last polished shape. Hands-on workouts support scholars comprehend an idea quickly after studying it. The textual content adopts a spiral process: many themes are revisited a number of occasions, occasionally from a special point of view or at a better point of complexity. The aim is to slowly strengthen scholars’ problem-solving and writing abilities.

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**Additional resources for A Spiral Workbook for Discrete Mathematics**

**Example text**

For each statement, (i) represent it as a formula, (ii) find the negation (in simplest form) of this formula, and (iii) express the negation in words. (a) For all real numbers x and y, x + y = y + x. (b) For every positive real number x there exists a real number y such that y 2 = x. (c) There exists a real number y such that, for every integer x, 2x2 + 1 > x2 y. 8. For each statement, (i) represent it as a formula, (ii) find the negation (in simplest form) of this formula, and (iii) express the negation in words.

4. Idempotent laws: When an operation is applied to a pair of identical logical statements, the result is the same logical statement. Compare this to the equation x2 = x, where x is a real number. It is true only when x = 0 or x = 1. But the logical equivalences p ∨ p ≡ p and p ∧ p ≡ p are true for all p. 5. De Morgan’s laws: When we negate a disjunction (respectively, a conjunction), we have to negate the two logical statements, and change the operation from disjunction to conjunction (respectively, from conjunction to a disjunction).

Each column can be filled with m/2 = t non-overlapping dominoes placed vertically. As a result, the entire chessboard can be covered with nt non-overlapping vertical dominoes. 4 Show that, between any two rational numbers a and b, where a < b, there exists another rational number. Hint: Try the midpoint of the interval [a, b]. 5 Show that, between any two rational numbers a and b, where a < b, there exists another rational number closer to b than to a. Hint: Use a weighted average of a and b. Sometimes a non-constructive proof can be used to show the existence of a certain quantity that satisfies some conditions.