1. Functional Notation Addendum to Chapter 4

2. 2 Logic Notation Systems We have seen three different, but equally powerful, notational systems for describing the behaviour of gates and circuits: Boolean expressions logic diagrams truth tables

3. 3 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.

4. 4 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.

5. 5 A Fourth System In addition to these three, there is another widely used system of notation for logic. Functional Notation

6. 6 … 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)

7. 7 Functional Equivalents Boolean NotationFunctional Notation X=A’X=NOT(A) X=A + BX=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))

8. 8 XOR XOR must be defined in terms of the 3 logic primitives: AND, OR, and NOT. Recall its explanation: “one or the other and not both”

9. 9 XOR This translates into Boolean Notation as follows: “one or the other” and not both X = (A + B)  (A  B) ’

10. 10 XOR The Boolean Notation X = (A + B)  (A  B) ’ translates as: X = AND( OR(A,B), NOT( AND(A,B))) in Functional Notation.

11. 11 XOR The truth table for XOR reveals a hint for simplifying our expression. ABXOR 000 011 101 110

12. 12 XOR Note that XOR is False (0) when A and B are the same, and True (1) when they are different. ABXOR 000 011 101 110

13. 13 XOR So XOR can be expressed very simply as: X=NOT(A=B) or X=A<>B

14. 14 XOR Notice 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.

15. 15 Consider this familiar circuit How can this circuit be expressed in Functional Notation?

16. 16 Equivalent expressions Recall the Boolean expression for the circuit: X=(AB + AC) Page 99

17. 17 Equivalent expressions Replace the most “internal” operators with functional expressions: X=(AB + AC) AND(A,B) AND(A,C)

18. 18 Equivalent expressions Now replace the “external” operators, working “outwards”: X=(AB + AC) X=OR(AND(A,B), AND(A,C))

19. 19 The equivalent circuit

20. 20 The equivalent circuit The Boolean expression: X = A  (B + C) X = AND(A, OR(B,C)) and its Functional equivalent.