By Harris Kwong
This can be a textual content that covers the traditional themes in a sophomore-level direction in discrete arithmetic: common sense, units, facts thoughts, uncomplicated quantity concept, capabilities, kinfolk, and straight forward combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in arithmetic, and publications the scholars via a close dialogue on how an evidence is revised from its draft to a last polished shape. Hands-on routines support scholars comprehend an idea quickly after studying it. The textual content adopts a spiral strategy: many subject matters are revisited a number of occasions, occasionally from a distinct point of view or at the next point of complexity. The objective is to slowly enhance scholars’ problem-solving and writing abilities.
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Additional resources for A Spiral Workbook for Discrete Mathematics
Summary and Review • A biconditional statement p ⇔ q is the combination of the two implications p ⇒ q and q ⇒ p. • The biconditional statement p ⇔ q is true when both p and q have the same truth value, and is false otherwise. • A biconditional statement is often used in defining a notation or a mathematical concept. 4 1. Let p, q, and r represent the following statements: p: q: r: Sam had pizza last night. Chris finished her homework. Pat watched the news this morning. Give a formula (using appropriate symbols) for each of these statements.
8. Find a counterexample to the following claim: For any positive integer n, if n is prime, then n2 + 4 is also prime. 2 47 Direct Proofs Direct Proofs To show that a statement q is true, follow these steps: 1. Either find a result that states p ⇒ q, or prove that p ⇒ q is true. 2. Show or verify that p is true. 3. Conclude that q must be true. The logic is valid because if p ⇒ q is true and p is true, then q must be true. Symbolically, we are saying that the logical formula [(p ⇒ q) ∧ p] ⇒ q is a tautology (we can easily verify this with a truth table).
Show that p is true (that is, show that p is false). The proof may proceed as follow: Proof: We want to prove the contrapositive of the stated result. Assume q is false, . . . 1 Remember to describe your goal. Therefore p is false. Let n be an integer. Show that if n2 is even, then n is also even. Solution: Proof by contrapositive: We want to prove that if n is odd, then n2 is odd. If n is odd, then n = 2t + 1 for some integer t. Hence, n2 = 4t2 + 4t + 1 = 2(2t2 + 2t) + 1 is odd. This completes the proof.
A Spiral Workbook for Discrete Mathematics by Harris Kwong