By Harris Kwong
This can be a textual content that covers the traditional issues in a sophomore-level path in discrete arithmetic: common sense, units, facts concepts, simple quantity idea, features, relatives, and undemanding combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in arithmetic, and publications 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 technique: many themes are revisited a number of occasions, occasionally from a distinct point of view or at a better point of complexity. The aim is to slowly improve scholars’ problem-solving and writing talents.
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Additional info for A Spiral Workbook for Discrete Mathematics
Let n be an integer. (a) Prove that if n is a multiple of 3, then n2 is also a multiple of 3. (b) Prove that if n is a multiple of 7, then n3 is also a multiple of 7. 6. Prove that if n is not a multiple of 3, then n2 is also not a multiple of 3. Hint: If n is not a multiple of 3, then n = 3q + 1 or n = 3q + 2 for some integer q. 7. Use the facts that √ (i) 2 is irrational, and √ (ii) if x is irrational, then x is also irrational, √ to prove that 8 2 is irrational. 8. Recall that we can use a counterexample to disprove an implication.
Derive (b) as a corollary of (a). (d) Show that if m and n are odd, then so is mn. (e) Show that if m is even, and n is odd, then mn is even. 4. Prove that, for any odd integer n, the number 2n2 + 5n + 4 must be odd. 5. Let n be an integer. (a) Prove that if n is a multiple of 3, then n2 is also a multiple of 3. (b) Prove that if n is a multiple of 7, then n3 is also a multiple of 7. 6. Prove that if n is not a multiple of 3, then n2 is also not a multiple of 3. Hint: If n is not a multiple of 3, then n = 3q + 1 or n = 3q + 2 for some integer q.
Explain. if false, must p be false? Explain. Biconditional Statements The biconditional statement “p if and only if q,” denoted p ⇔ q, is true when both p and q carry the same truth value, and is false otherwise. ” Its truth table is depicted below. 4 25 Biconditional Statements are true, because, in both examples, the two statements joined by ⇔ are true or false simultaneously. A biconditional statement can also be defined as the compound statement (p ⇒ q) ∧ (q ⇒ p). This explains why we call it a biconditional statement.
A Spiral Workbook for Discrete Mathematics by Harris Kwong