Page 45 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes 0 + 1 + 1 = 1
1 + 1 + 1 = 1
0 + 1 + 1 + 1 = 1

1 + 0 + 1 + 1 + 1 = 1
Take a close look at the two-term sums in the first set of equations. Does that pattern look familiar
to you? It should! It is the same pattern of 1’s and 0’s as seen in the truth table for an OR gate.
In other words, Boolean addition corresponds to the logical function of an “OR” gate, as shown
in Figure 3.1
Figure 3.1: Boolean Addition and the OR Gate

There is no such thing as subtraction in the realm of Boolean mathematics. Subtraction implies
the existence of negative numbers: 5 – 3 is the same thing as 5 + (–3), and in Boolean algebra
negative quantities are forbidden. There is no such thing as division in Boolean mathematics,
either, since division is really nothing more than compounded subtraction, in the same way that
multiplication is compounded addition.
Multiplication is valid in Boolean algebra, and thankfully it is the same as in real-number algebra:
anything multiplied by 0 is 0, and anything multiplied by 1 remains unchanged:
0 × 0 = 0

0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
This set of equations should also look familiar to you: it is the same pattern found in the truth
table for an AND gate. In other words, Boolean multiplication corresponds to the logical function
of an “AND” gate.

Like “normal” algebra, Boolean algebra uses alphabetical letters to denote variables. Unlike
“normal” algebra, though, Boolean variables are always UPPERCASE letters, never lower-case.
Because they are allowed to possess only one of two possible values, either 1 or 0, each and every
variable has a complement: the opposite of its value. For example, if variable “A” has a value
of 0, then the complement of A has a value of 1. Boolean notation uses a bar above the variable
character to denote complementation, like this.
If: A = 0
Then: A = 1
If: A = 1
Then: A = 0

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