By Harris Kwong

It is a textual content that covers the traditional subject matters in a sophomore-level path in discrete arithmetic: good judgment, units, facts thoughts, easy quantity idea, features, kinfolk, and easy 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 workouts support scholars comprehend an idea quickly after studying it. The textual content adopts a spiral technique: many themes are revisited a number of instances, occasionally from a special viewpoint or at a better point of complexity. The target is to slowly improve scholars’ problem-solving and writing talents.

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**Example text**

A proposition that is always false is called a contradiction. A proposition that is neither a tautology nor a contradiction is called a contingency . 1 From the following truth table p T F p F T p∨p T T p∧p F F we gather that p ∨ p is a tautology, and p ∧ p is a contradiction. In words, p ∨ p says that either the statement p is true, or the statement p is true (that is, p is false). This claim is always true. The compound statement p ∧ p claims that p is true, and at the same time, p is also true (which means p is false).

Therefore, instead of proving p ⇒ q, we may prove its contrapositive q ⇒ p. Since it is an implication, we could use a direct proof: Proof by contrapositive. 1. Assume q is true (hence, assume q is false). 2. 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.

B) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). Solution: The truth tables for (a) and (b) are depicted below. p F F T T p∨q T F T T p ∧ (q ∨ r) p ∧ q T T T T T F F F F F F F F F F F q∧r T F T F F F F F p T T F F p T T T T F F F F q T T F F T T F F r T F T F T F T F q∨r T T T F T T T T q T F T F p⇒q T F T T (p ∧ q) ∨ (p ∧ r) T T T F F F F F Example (a) is an important result. It says that p ⇒ q is true when one of these two things happen: (i) when p is false, (ii) otherwise (when p is true) q must be true.