By Harris Kwong

This can be a textual content that covers the normal issues in a sophomore-level path in discrete arithmetic: good judgment, units, evidence concepts, easy quantity conception, features, family, and trouble-free 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 evidence is revised from its draft to a last polished shape. Hands-on routines support scholars comprehend an idea quickly after studying it. The textual content adopts a spiral strategy: many subject matters are revisited a number of instances, occasionally from a distinct viewpoint or at the next point of complexity. The aim is to slowly enhance scholars’ problem-solving and writing talents.

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Verify their compositeness by means of factorization. 4 Let m and n be positive integers. Show that, if mn is even, then an m × n chessboard can be fully covered by non-overlapping dominoes. Remark. This time, the names m and n have already been assigned to the two positive integers. 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.

3. Distributive laws: When we mix two different operations on three logical statements, one of them has to work on a pair of statements first, forming an “inner” operation. This is followed by the “outer” operation to complete the compound statement. Distributive laws say that we can distribute the “outer” operation over the inner one. 4. Idempotent laws: When an operation is applied to a pair of identical logical statements, the result is the same logical statement. Compare this to the equation x2 = x, where x is a real number.

7. Identity laws: Compare them to the equation x · 1 = x: the value of x is unchanged after multiplying by 1. We call the number 1 the multiplicative identity. For logical operations, the identity for disjunction is F, and the identity for conjunction is T. 8. Domination laws: Compare them to the equation x · 0 = 0 for real numbers: the result is always 0, regardless of the value x. The “zero” for disjunction is T, and the “zero” for conjunction is F. 6 What is the negation of 2 ≤ x ≤ 3? Give a logical explanation as well as a graphical explanation.

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