Chapter 03 - Boolean Algebra II-2x2(1)

# Similarly some of the theorems of ordinary algebra

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Unformatted text preview: y compare both sides of the equation and let the different between them serve as a guide for what steps to take next. Example 1 (p. 71) Differences between Boolean algebra and ordinary algebra As we have previously observed, some of the theorems of Boolean algebra are not true for ordinary algebra. Similarly, some of the theorems of ordinary algebra are not true for Boolean algebra. Consider, for example, the cancellation law for ordinary algebra: If x + y = x + z, then y = z (3-31) The cancellation law is not true for Boolean algebra. We will demonstrate this by constructing a counterexample in which x + y = x + z but y ≠ z. Let x = 1, y = 0, z = 1. Then, 1 + 0 = 1 + 1 but 0 ≠ 1 Section 3.5 (p 72) Similarities between Boolean algebra and ordinary algebra Even though the statements in the previous 2 slides (331 and 3-32) are generally false for Boolean algebra, the converses are true: If y = z, then x + y = x + z (3-33) If y = z, then xy = xz (3-34) Section 3.5 (p 72) Ordinary algebra: the cancellation law for multiplication is If xy = xz, then y = z (3-32) This law is valid provided x ≠ 0 In Boolean algebra, the cancellation law for multiplication is also not valid when x = 0. Let x = 0, y = 0, z = 1; then 0 • 0 = 0 • 1, but 0 ≠ 1 Because x = 0 about half the time in switching algebra, the cancellation law for multiplication cannot be used....
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