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)

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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.