By Harris Kwong

It is a textual content that covers the traditional issues in a sophomore-level direction in discrete arithmetic: common sense, units, facts innovations, easy quantity conception, services, kinfolk, and effortless 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 method: many issues are revisited a number of instances, occasionally from a special point of view or at the next point of complexity. The target is to slowly strengthen scholars’ problem-solving and writing talents.

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**Example text**

3. Let n be an integer. Prove that if n is even, then n2 = 4s for some integer s. 4. Let m and n be integers. Show that mn = 1 implies that m = 1 or m = −1. √ 5. Let x be a real number. Prove√by contrapositive: if x is irrational, then √x is irrational. Apply this result to show that 4 2 is irrational, using the assumption that 2 is irrational. 6. Let x and y be real numbers such that x = 0. Prove that if x is rational, and y is irrational, then xy is irrational. √ 7. Prove that 5 is irrational.

Dogs are mammals. Mammals are vertebrates. German shepherds are vertebrates. is valid because of the law of syllogism. The big question is, how can we prove an implication? The most basic approach is the direct proof : 1. Assume p is true. 2. Deduce from p that q is true. Proving p ⇒ q via a direct proof. The important thing to remember is: use the information derived from p to show that q is true. This is how a typical direct proof may look: Proof: Assume p is true. Then . . Because of p, we find .

2 (Intermediate Value Theorem) Let f be a function that is continuous over a closed interval [a, b]. Then f assumes all values between f (a) and f (b). In other words, for any value t between f (a) and f (b), there exists a number c inside [a, b] such that f (c) = t. Both results only guarantee the existence of a number c with some specific property; they do not tell us how to find this number c. Nevertheless, the Mean-Value Theorem plays a very important role in analysis; many of its applications are beyond the scope of this course.