Example of Digital Problem

1. Example of Digital Problem ð To build a logic circuit of an Alarm System at the office (with one door and one window) that will ring if the door or window is/are opened after working hours. ð Followings are the steps that are to be taken to build a logical circuit. • Problem Determination -- Determine the problem that has to be solved -- A logical circuit that will trigger the emergency bell if door or window is/are opened outside office hours has to be built 2. Conceptualization - Obtain the relevant logical variable and make a logical table and also a truth table. Obtain the logical expression from the truth table

2. The related variables are : Time T = 0 (work time) T = 1 (not work time) Doors D = 0 (closed) D = 1 (opened) Windows W = 0 (closed) W = 1 (opened) • Whether Bell B will ring (1) or will not ring (0) depends on all three logical variables (depending on the condition or problem given)

3. INPUT VARIABLES OUTPUT Time Door Window Bell Work closed closed Doesn’t ring Work closed opened Doesn’t ring Work opened closed Doesn’t ring Work opened opened Doesn’t ring Not Work closed closed Doesn’t ring Not Work closed opened Will Ring Not Work opened closed Will Ring Not Work opened opened Will Ring Logic Table

4. INPUTS OUTPUT T D W B 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Truth Table (based on logical variable and function above)

5. WD 00 01 11 10 T 0 1 1 1 1 TW TD • B is the output or the function that we have to find. The B function or expression can be obtained through many was based on the truth table above. One of the methods is by using a Karnaugh map. Below is the Karnaugh Map for the Truth Table above From the Karnaugh Map above, B = TW + TD • The expression B above can be further summarized using theorem or the laws in Boolean algebra • B = TW + TD = T(W + D) (from the Law of Distribution)

6. T T(W + D) W (W + D) D • From the expression obtained, the logical circuit for B=T(W + D) is as follow :

7. Building logical circuits using only NAND or only NOR gates • Most components in computers are built using only either the NAND or the NOR. This can further simplify the construction of such circuits (i.e. do not need to use various gates in a logic circuit) • To build a circuit that only uses NAND or NOR gates, firstly the expression for the circuit has to be changed into an expression that only has either the NAND or NOR operations. To change it, the De Morgan and Involution Theorems are used.

8. A = A The Involution Theorem is as follow: Example: Take the B expression B = T(W + D) as an example. 1. Using only the NAND gate • To get an expression that only uses the NAND gate, eliminate the OR operation in the expression by using the Involution Theorem and De Morgan theorem.

9. Example : B = T.(W + D) = T.(W + D) Involution Theorem = T. (W . D) De Morgan theorem = T. (W . D) Involution Theorem • Hence, the logic circuit for B that only uses the NAND gate can be drawn as follow:

10. T W D

11. 2. Using only the NOR gate • To get an expression that only uses the NOR gate, eliminate the AND operation in the expression by using the Involution Theorem and the De Morgan Theorem. Example: B = T.(W + D) = T . (W + D) Involution Theorem = T + (W + D) De Morgan theorem

12. T W D • It is found that the expression produced no longer has the AND operation and all the OR operation has the complement sign or NOT symbol (or the NOR operation). Hence, the logic circuit for L that only uses the NOR gate can be drawn as follow:

13. Circuits from NAND and NOR gates • Buying separate AND, OR gates etc. is wasteful • These gates can be made of NAND or NOR gates • NAND gates are particularly cheap • Sometimes there can be redundancy, so circuits can be simplified