By Harris Kwong
It is a textual content that covers the normal issues in a sophomore-level path in discrete arithmetic: common sense, units, evidence suggestions, simple quantity concept, services, family, and effortless 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 evidence is revised from its draft to a last polished shape. Hands-on workouts aid 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 special point of view or at a better point of complexity. The aim is to slowly enhance scholars’ problem-solving and writing abilities.
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Additional resources for A Spiral Workbook for Discrete Mathematics
So we may assume n is not divisible by 3. Then what would n look like, and, what can you say about n + 2 and n + 4? 3. Let n be an integer. (a) Show that if n is odd, then n2 is also odd. (b) Show that if n is odd, then n4 is also odd. (c) A corollary is a result that can be derived easily from another result. 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.
To evaluate yz −3 , we have to perform exponentiation first. Hence, yz −3 = y · z −3 = zy3 . 3 3 Another example: the notation x2 means x raised to the power of 23 , hence x2 = x8 ; it should not be interpreted as (x2 )3 , because (x2 )3 = x6 . 5 It is not true that p ⇔ q can be written as “p ⇒ q ∧ q ⇒ p,” because it would mean, technically, p ⇒ (q ∧ q) ⇒ p. The correct notation is (p ⇒ q) ∧ (q ⇒ p). 2 Insert parentheses in the following formula p⇒q∧r to identify the proper procedure for evaluating its truth value.
Since x is rational, we also have x= p q for some integers p and q, where q = 0. It follows that m p = x + y = + y. 3 55 Indirect Proofs where mq − np and nq are both integers, with nq = 0. This makes y rational, which contradicts the assumption that y is irrational. Thus, x + y cannot be rational, it must be irrational. 4 Prove that √ √ √ x+y = x+ y for any positive real numbers x and y. Hint. The words “for any” suggest this is a universal quantification. Be sure you negate the problem statement properly.
A Spiral Workbook for Discrete Mathematics by Harris Kwong