functional notation addendum to chapter 4. 2 logic notation systems we have seen three different,...
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Functional Notation
Addendum to Chapter 4
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Logic Notation SystemsWe have seen three different, but equally powerful, notational systems for describing the behaviour of gates and circuits:
Boolean expressions logic diagrams truth tables
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Recall that…Boolean expressions: expressions in Boolean algebra, a mathematical notation for expressing two-valued logic.
This algebraic notation is an elegant and powerful way to demonstrate the activity of electrical circuits.
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Recall further that…Logic diagram: A graphical representation of a circuit.
Each type of gate is represented by a specific graphical symbol.
Truth table: A table showing all possible input values and the output values associated with each set of inputs.
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A Fourth SystemIn addition to these three, there is another widely
used system of notation for logic.
Functional Notation
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Functional Notation… uses a function name followed by a list of arguments in place of the operators used in Boolean Notation.
For example:
A’ becomes NOT(A)
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Functional EquivalentsBoolean Notation Functional Notation
X=A’ X=NOT(A)
X=A + B X=OR(A,B)
X=A B X=AND(A,B)
X=(A + B)’ X=NOT(OR(A,B))
X=(A B)’ X=NOT(AND(A,B))
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XORXOR must be defined in terms of the 3 logic
primitives: AND, OR, and NOT.
Recall its explanation:
“one or the other and not both”
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XORThis translates into Boolean Notation as follows:
“one or the other” and not both
X = (A + B) (A B) ’
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XORThe Boolean Notation
X = (A + B) (A B) ’
translates as:
X = AND( OR(A,B), NOT( AND(A,B)))in Functional Notation.
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XORThe truth table for XOR
reveals a hint for simplifying our expression.
A B XOR0 0 0
0 1 1
1 0 1
1 1 0
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XORNote that XOR is False (0)
when A and B are the same, and True (1) when they are different.
A B XOR0 0 0
0 1 1
1 0 1
1 1 0
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XORSo XOR can be expressed very simply as:
X=NOT(A=B)or
X=A<>B
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XORNotice that this expression is not
strictly functional since it uses the ‘not equal’ operator.
However, we’re more interested in implementing logic in Excel, than strict Functional Notation.
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Consider this familiar circuit
How can this circuit be expressed in Functional Notation?
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Equivalent expressionsRecall the Boolean expression for the
circuit:
X=(AB + AC)
Page 99
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Equivalent expressionsReplace the most “internal” operators
with functional expressions:
X=(AB + AC)
AND(A,B) AND(A,C)
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Equivalent expressionsNow replace the “external” operators,
working “outwards”:
X=(AB + AC)
X=OR(AND(A,B), AND(A,C))
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The equivalent circuit
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The equivalent circuitThe Boolean expression:
X = A (B + C)
X = AND(A, OR(B,C))
and its Functional equivalent.