1. BOOLEAN ALGEBRA LOGIC GATES

2. Introduction British mathematician George Boole(1815-1864) was successful in finding the link between logic and mathematics. Boolean Algebra is fundamentally important in the design of circuits used in computers.

3. Boolean Algebra In Boolean Algebra, elements have one of two values –True or False. The circuits in a computer are also designed for two-state operations. That is input and output of a circuit is either low(0) or high(1). The circuits are called logic circuits.

4. BOOLEAN VARIABLES The variables in Boolean Algebra can take only two values- True or false. The variables are called Boolean variables.

5. Boolean Operators There are three basic operators in Boolean Algebra which are called logical operators or Boolean operators. OR - logical addition AND – logical multiplication NOT – Logical negation The Boolean operators are used to combine Boolean variables and Boolean constants to form Boolean Expressions.

6. TRUTH TABLES A truth table is a table that represents the possible values of the operands and corresponding values of a Boolean operation or a Boolean expressions. Boolean expression with ‘n’ number of variables, the truth table will have 2 n rows.

7. LOGIC GATES A logic gate is an electronic circuit which makes logic decisions. A logic gate takes one or more inputs and will produce only one output. Logic gates are the building blocks from which most of the digital systems are built up.

8. The three basic logical operators are OR,AND and NOT They are said to be logically complete as any Boolean function can be realized in terms of these connectives. Gate used to implement these logical operators are known as basic logic gates.

9. OR Operation Boolean expression for the OR operation: x =A + B The above expression is read as “x equals A OR B”

10. OR Gate An OR gate is a gate that has two or more inputs and whose output is equal to the OR combination of the inputs.

11. AND Gate The and gate is a logic circuit which accepts two or more input signals but produces only one output signal. The output signal produced will be 1 only if all input signals are 1 otherwise it will be 0.

12. AND Operation Boolean expression for the AND operation: x =A B The above expression is read as “x equals A AND B”

13. AND Gate An AND gate is a gate that has two or more inputs and whose output is equal to the AND product of the inputs.

14. NOT Gate The NOT gate is a logic circuit which will accept only one input signal. The output state produced by NOT gate will be always the opposite of the input signal Hence it is called the inverter.

15. NOT Operation The NOT operation is an unary operation, taking only one input variable. Boolean expression for the NOT operation: x = A The above expression is read as “x equals the inverse of A Also known as inversion or complementation. Can also be expressed as: A’

17. Boolean Theorems (Single-Variable) ‏ x* 0 =0 x* 1 =x x*x=x x*x’=0 x+0=x x+1=1 x+x=x x+x’=1

18. X Y=YX X+(Y+Z)=(X+Y)+Z X(YZ)=(XY)Z X(Y+Z)=XY+XZ (X’)’=X Boolean theorems (multivariable)

19. DeMorgans law 1 (A.B)’= A’+ B’ 2 (A+B)’= A’. B’

20. Chapter 3: Digital Logic 20 DeMorgan’s law can be extended to any number of variables. Replace each variable by its complement and change all AND to OR and all OR to AND. Thus, we find the complement of: is:

21. ADVANCED GATES

22. NAND Gate The name NAND is the short form of “NOT AND”. As the name implies NAND gate can be formed by inverting the output of AND gate. NAND gate can be represented by using AND and NOT gate

23. NAND Gate Boolean expression for the NAND operation: x = A B

24. NOR Gate The NOR gate is the short form of “NOT OR”. As the name implies NOR gate can be implemented by inverting the output of OR gate.

25. Logic symbol

26. UNIVERSAL GATES The NAND and NOR gates can be easily used to implement all the basic logic gates such as AND,OR and NOT. Actually NAND and NOR gates are more popular as they are less expensive and easier to design. Due to their versatility they are often referred to as Universal gates.

27. Universality of NAND Gates

28. Universality of NOR Gates

29. XOR Gate The exclusive operation produces a high logical output when the two inputs are at opposite logic levels. Note the special symbol  for the XOR operation.

30. Boolean Algebra Through our exercises in simplifying Boolean expressions, we see that there are numerous ways of stating the same Boolean expression. –These “synonymous” forms are logically equivalent. –Logically equivalent expressions have identical truth tables. In order to eliminate as much confusion as possible, designers express Boolean functions in standardized or canonical form.

31. Boolean Algebra There are two canonical forms for Boolean expressions: sum-of-products and product-of- sums. –Recall the Boolean product is the AND operation and the Boolean sum is the OR operation. In the sum-of-products form, ANDed variables are ORed together. –For example: In the product-of-sums form, ORed variables are ANDed together: –For example:

32. Boolean Algebra It is easy to convert a function to sum-of-products form using its truth table. We are interested in the values of the variables that make the function true (=1). Using the truth table, we list the values of the variables that result in a true function value. Each group of variables is then ORed together.

33. Boolean Algebra The sum-of-products form for our function is: We note that this function is not in simplest terms. Our aim is only to rewrite our function in canonical sum-of-products form.

34. MAP SIMPLIFICATION

35. Two methods of simplifying algebraic expressions are the map method and tabular method. The map method is used for functions upto six variable. To manipulate functions of a large number of variables, the tabular method is also known as the Quine-McCluskey method is used. The map method is also known as the karnaugh map or k-map.

36. Each combination of the variable in a truth table is called a minterm.When expressed in a truth table a function of n variable will have 2 n minterms.

37. A Boolean function represented by a truth table is plotted into the map by inserting 1’s in those squares where the function is 1. The squares containing 1’s are combined groups of adjacent squares that is an integral power of 2. Groups of combined adjacent squares with one or more groups. Each group of squares represents an algebraic term, and or of those terms terms gives the simplified algebraic expression for the function.

38. Two variable k-map 0 1 2 3 A B 0 B 1 A 1

39. Three variable k-map 0 1 3 2 4 5 7 6 A BC 00 01 1110 0 1 B A C

40. 4 VARIABLE K-MAP 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 AB CD 00 01 11 10 00 01 1110 A D B C

41. F= (3,4,6,7) Simplify the Boolean function using map method. 1 1 1 1 1 1 1 A C B

42. There are four squares marked with 1’s,one for each minterm that produces 1 for the function. Two adjacent squares are combined in the third column.the column belongs to both B and C, and produces the term BC. The remaining square produce term AC’. F=BC+AC’

43. Product of sum simplification 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1

44. If the squares marked with o’s are combined, as shown in the diagram we obtain the given formula. F’=AB+CD+BD’ Taking the complement of f’, we obtain the simplified function in product-of-sums form F=(A’+B’)(C’+D’)(B’+D)