1. Lecture 2 – Boolean Algebra Lecturer: Amy Ching Date: 21 st Oct 2002

2. Binary Systems Computer hardware works with binary numbers, but binary arithmetic is much older than computers Ancient Chinese Civilisation (3000 BC) Ancient Greek Civilisation (1000 BC) Boolean Algebra (1850)

3. Propositional Logic The Ancient Greek philosophers created a system to formalise arguments called propositional logic. A proposition is a statement that could be TRUE or FALSE Propositions could be compounded into by means of the operators AND, OR and NOT

4. Propositional Calculus Example Propositions, that may be TRUE or FALSE: it is raining the weather forecast is bad A combined proposition: it is raining OR the weather forecast is bad

5. Propositional Calculus Example We can equate propositions, for example by writing: I will take an umbrella = it is raining OR the weather forecast is bad or equivalently we can write: If it is raining OR the weather forecast is bad Then I will take an umbrella OR Rain Bad Weather Forecast Take Umbrella

6. Diagrammatic representation We can think of the umbrella proposition as a result that we calculate from the weather forecast or the fact that it is raining Rain Bad Weather Forecast OR Take Umbrella

7. Truth Tables Since propositions can only take two values, we can express all possible outcomes of the umbrella proposition by a table: RainingBad WeatherUmbrella False True FalseTrue

8. Boolean Algebra Propositional logic is too cumbersome to express arguments of any complexity. An equivalent, more tractable system of logic was introduced by the English mathematician Boole in 1850.

9. Boolean Algebra A Boolean variable has only one of two values: true or false (1 or 0), called logic values. A Boolean variable can be a function of other Boolean variables, i.e. Z = F(A, B, C, D … ). We can also express the function in terms of a Truth Table A Truth Table is a tabulated list contains a clear relationship between all possible combination of input variables and the resultant operation.

10. Fundamental Operators And Operator Three fundamental operators AND, OR and NOT. AND Operator Z = A  B The AND operation is represented by the symbol “  ”. The truth table or logic table of the AND operation is as follows:

11. Fundamental Operators – OR Operator OR Operator Z = A + B The OR operation is represented by the “ + ” symbol. Note that the OR operation is not related to addition in ordinary arithmetic. The truth table for OR is as follows:

12. Fundamental Operators – NOT Operator NOT Operator or Z = A ’ The NOT operation is designated by an overline or an hyphen. In words, the above expression is “ Z ” is equal to a NOT ”. The truth table for the NOT operation is as follows: The NOT operation is a complement operation.

13. Fundamentals of Boolean Algebra The truth values are replaced by 1 and 0: 1 = TRUE 0 = FALSE Operators are replaced by symbols ' = NOT + = OR = AND

14. Precedence Further simplification is introduced by introducing a precedence for the evaluation of the operators. (The highest precedence operator is evaluated first.) OperatorSymbolPrecedence NOT ' Highest AND Middle OR + Lowest

15. All outcomes can be written as: AND Operator ( ) OR Operator (+) NOT '

16. Boolean Algebra Laws 1) Communicative laws2) Associative laws A + B = B + A A+(B+C) = (A+B)+C AB = BA (AB)C = A(BC) 3) Distributive laws4) Absorption Law A  (B+C) = (A  B) + (A  C) A  (A+B) = A +(A  B) 5) Complement Law A + = 1 A  = 0 Other useful relationship: 1) A  1 = A2) A  0 = 0 3) A + 1 = 14) A + 0 = A 5) A + A = A6) A  A = A

17. DeMorgan ’ s Theorem 1) 2) Both forms of the DeMorgan ’ s Theorem have complement of an entire expression, and the effect of this complementing is to interchange each “ + ” to a “  ” and each “  ” to a “ + ” and to complement each variable Expression 1) is also described as inputs A and B with a NAND operator Expression 2) is also described as inputs A and B with a NOR operator

18. Simplification of Boolean Equation Using DeMorgan ’ s Theorem Simplify Y = (A+B)  (A+C) Y = (A+B)  (A+C) = A  A + A  C + B  A + B  C = A + A  C + A  B + B  C = A  (1+C+B) + B  C – Redundance Law = A + B  C

19. Sum of Product (SOP) and Product of Sum (POS) Product term - is a single variable of the logic product of several variables. The variables may or may not be complemented. e.g. XYZ, Y Sum term - is a single variable or the sum of several variables. The variable may or may not be complemented e.g. X+Y, Sum of products expression - is a product term of several product terms logically added together e.g. Product of sums expression - is a sum term or several sum terms logically multiplied together e.g.

20. Conversion of a truth table into SOP and POS Sum of product solution Product of sum solution

21. Derivation of SOP and POS Sum of Products expression (Minterm Form) 1) From a truth table 2) The product terms from each row in which the output is a “ 1 ” are collected 3) The desired expression is the sum of these products e.g. Product of Sums expression (Maxterm Form) 1) Form a truth table 2) Construct a column to contain the sum terms 3) The required expression is the product of sums terms from the row in which the output is “ 0 ” e.g.

22. Karnaugh Map (K-Map) The Karnaugh map provides a formal systematic approach to the problem of minimisation of logic functions. e.g. In the Karnaugh map, every possible combination of the binary input variables is represented on the map by a square ( or cell). For N input variables, we have 2 n square.

23. Layout of Karnaugh Map

24. Use of K-Map In this way, by inspection, it is obvious that terms can be combined and simplified using the theorem. e.g. To plot the SOP function on Karnaugh map, a “ 1 ” is entered in each square corresponding to a product term in the function.

25. Use of K-Map To use the map to form the POS function, a “ 0 ” is entered in each cell corresponding to each sum term in the function. Result of simplification should then be in POS form.

26. Representation of Karnaugh Map Truth Table vs Karnaugh Map Truth TableKarnaugh Map

27. Use of K-Map There is a correspondence between top and bottom rows, and between extreme left and right-hand columns.

28. Simplification using a K-Map Simplify Solution

29. Example 1

30. Example 2

31. Example 3

32. Example 4

33. Example 5