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 (331)
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 332) are generally false for Boolean algebra,
the converses are true: If y = z, then x + y = x + z (333)
If y = z, then xy = xz (334) Section 3.5 (p 72) Ordinary algebra: the cancellation law for multiplication is
If xy = xz, then y = z (332)
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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 Spring '08
 Brown

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