By Harris Kwong
It is a textual content that covers the normal issues in a sophomore-level direction in discrete arithmetic: common sense, units, evidence options, easy quantity concept, capabilities, kin, and undemanding 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 aid scholars comprehend an idea quickly after studying it. The textual content adopts a spiral technique: many issues are revisited a number of occasions, occasionally from a unique standpoint or at a better point of complexity. The objective is to slowly enhance scholars’ problem-solving and writing abilities.
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Additional info for A Spiral Workbook for Discrete Mathematics
Thus, we can refer to them in the proof without an introduction. Solution: Since mn is even, one of the two integers m and n must be even. Without loss of generality (since the other case is similar), we may assume m, the number of rows, is even. Then m = 2t for some integer t. Each column can be filled with m/2 = t non-overlapping dominoes placed vertically. As a result, the entire chessboard can be covered with nt non-overlapping vertical dominoes. 4 Show that, between any two rational numbers a and b, where a < b, there exists another rational number.
This counterexample shows that the second statement is false. 10 Consider True or false: ∃y ∈ R ∀x ∈ Z (xy < 1)? Many theorems in mathematics can be expressed as quantified statements. ” The keyword “whenever” suggests that we should use a universal quantifier. ∀x, y (x is rational ∧ y is irrational ⇒ x + y is irrational). It can also be written as ∀x ∈ Q ∀y ∈ / Q (x + y is irrational). Although this form looks complicated and seems difficult to understand (primarily because it is quite symbolic, hence appears to be abstract and incomprehensible to many students), it provides an easy form for negation.
If x is in S, then x2 is in S. Therefore, if x2 is in S, then x is in S. 3 Proof Techniques Indirect Proofs Instead of proving p ⇒ q directly, it is sometimes easier to prove it indirectly. There are two kinds of indirect proofs: the proof by contrapositive, and the proof by contradiction. The proof by contrapositive is based on the fact that an implication is equivalent to its contrapositive. 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.