Boolean Algebra & De Morgan's Theorems

1. Boolean Algebra & De Morgan's Theorems • Topics covered in this presentation: • Boolean Operations • Simplification of Boolean Expressions • De Morgan's Theorems

2. Boolean Algebra Boolean Algebra is a form of mathematics that can be used to help simplify a logic system. The inputs and outputs are variables and are represented by capital letters. For example, A and B may be used to represent inputs and Q to represent the output. In a digital system, each variable can have one of two states, either logic 1 or logic 0. As with ordinary algebra, the value of a variable can be defined. For example, A=1 would mean that variable A was at logic 1. All the basic logic functions can be represented in Boolean algebra, and a number of these can be grouped together to show the relationship between the inputs and the output.

3. Boolean Operations - AND The AND logic function will output a logic 1 when all inputs are at logic 1. In Boolean algebra, the AND function is indicated by a dot between the variables being ANDed together. A 2-input AND gate will be expressed as: Q = A.B Note that it does not matter in which order variables are ANDed together, A.B = B.A

4. A.A = 0 If an input is ANDed with an inverted version of itself, the output will always be equal to 0. Boolean Operations - AND The following are a set of rules for the AND logic function. All of these can be determined by examining the truth table for the AND function. A.1 = A If input A is ANDed with a logic 1 input, the output will always be equal to the value on input A. A.0 = 0 If input A is ANDed with a logic 0 input, the output will always be equal to 0. A.A = A If two identical inputs are ANDed together, the output will always be equal to the input value.

5. Boolean Operations - OR The OR logic function will output a logic 1 when at least one input is at logic 1. In Boolean algebra, the OR function is indicated by a plus sign between the variables being ORed together. A 2-input OR gate will be expressed as: Q = A+B Note that it does not matter in which order variables are ORed together. A+B = B+A

6. A.A = 1 If an input is ORed with an inverted version of itself, the output will always be equal to 1. Boolean Operations - OR The following are a set of rules for the OR logic function. All of these can be determined by examining the truth table for the OR function. A+1 = 1 If input A is ORed with a logic 1 input, the output will always be equal to 1. A+0 = A If input A is ORed with a logic 0 input, the output will always be equal to the value on input A. A+A = A If two identical inputs are ORed together, the output will always be equal to the input value.

7. In the case of the inverter, Q = A. A double bar above a variable indicates that the variable is inverted twice, therefore returning the variable to the original input value, A = A. Boolean Operations - NOT (Inversion) The NOT logic function has the effect of inverting the input and is therefore known as an inverter. In Boolean algebra, the function of inversion is indicated by a bar above the variable.

8. When an AND operation is inverted it becomes NAND: Q = A.B When an OR operation is inverted it becomes NOR: Q = A+B Inversion of AND and OR The function of inversion can be applied to operations as well as single variables. A bar above an AND or OR operation, indicates that the output is inverted.

9. Combinations of Logic Functions When more than one logic function is used, the order in which the logic functions are to be performed must be defined. As with ordinary algebra, Boolean equations can include brackets to define the order in which the operations are performed. The logic circuit opposite contains an OR and AND logic function connected together. This could be written as Q = A+B.C, but this doesn't indicate the order that the AND and the OR operation should be performed. Inputs A and B are first ORed together, and the output of the OR function is then ANDed with input C. The Boolean equation for this circuit is therefore: Q = (A+B).C

10. Simplifying Circuits with Boolean Algebra One of the reasons for using Boolean algebra is that it can be used to simplify a logic circuit by removing redundant terms to produce a circuit that requires fewer logic functions. When manipulating a Boolean expression the AND logic function can be treated in the same way as the mathematical operation of multiplication. The OR logic function can be treated in the same way as the mathematical operation of addition Sometimes it is necessary to expand equations in order to simplify them further.

11. Manipulation with Boolean Algebra In the same way as it is possible to take out a common multiplication factor in ordinary mathematics, the same can be carried out in Boolean algebra. In the example here, A is a common factor that can be taken out. A.B + A.C = A.(B + C) It is also possible to expand functions, such as the example below, using standard mathematical rules: A.(A + B) = A.A + A.B Note that since A.A = 0, this equation can be further simplified to get: A.A + A.B = 0 + A.B = A.B

12. Manipulation with Boolean Algebra Two logic identities used in simplification involve multiple variables: A + (A.B) = A A.(A + B) = A Both of these identities can be proven using Boolean algebra and the basic single-variable identities that have already been covered: A+(A.B) = (A.1) + (A.B) since A = A.1 = A. (1 + B) since A is a common factor = A since (1 + B) = 1 A.(A + B) = (A.A) + (A.B) expand terms = A + (A.B) since A.A = A = A since A + (A.B) = A

13. Q = A.(A +B).(A + B) Since A.(A + B) = A this can be simplified to: Q = A.(A + B) The terms can then be expanded to give: Q = (A.A) + (A.B) Since A.A = 0 this can be simplified to: Q = 0 + (A.B) = A.B Manipulation with Boolean Algebra Example Use Boolean algebra to simplify the logic circuit opposite. By inspecting the circuit it can be found that its logic function is: The simplified version of the above circuit is therefore:

14. De Morgan's first theorem can be stated as: A + B = A . B De Morgan's second theorem can be stated as: A . B = A + B De Morgan’s Theorems De Morgan's Theorems are used to help simplify a Boolean expression and can be used to eliminate ‘long bars’ that cover several variables. These theorems can be remembered by the phrase 'break the bar, and change the sign'.

15. Example Simplify the Boolean equation so that only single variables remain. Q = (A + B + C) . (A + B + C) Break the long bar, and change the sign linking the two parts. Q = (A + B + C) + (A + B + C) Break the bars and change the signs between the variables. Q = (A . B . C) + (A . B . C) Finally, since A = A, any double bars can be removed. Q = (A . B . C) + (A . B . C) De Morgan’s Theorem One use for De Morgan’s Theorems is to simplify Boolean equations that contain inversion bars across several variables, so that only single variable inversions remain.

16. Example Simplify the Boolean equation to use only NAND gates. Q = (A + B + C) . (A + B + C) Add a double bar above each half of the equation. This has no effect on the meaning of the equation but will help us to simplify it. Q = (A + B + C) . (A + B + C) Break a bar and change the sign between the variables. Q = (A . B . C) . (A . B . C) Finally, since A = A, any double bars can be removed. Q = (A . B . C) . (A . B . C) De Morgan’s Theorem The other use for De Morgan’s theorem is to convert an expression so that it only contains a single type of logic function.