BOOLEAN ALGEBRA INVESTIGATIONS

Transcription

1 OOLEN LGER INVESTIGTIONS NMES: dapted from If one of the inputs is always 1 (true), what is the output? 1 Therefore, + 1 = If one of the inputs is always 0 (false), what is the output? What variable is the output dependent upon? Therefore, + 0 = 3. If = 0, then the output is 0 If = 1, then the output is 1 How does the output compare with the input? It is the input Therefore, + = 4. If = 1, then the output is 1 If = 0, then the output is 1 How does the output compare with the input? lways True Therefore, + = If one of the inputs is always 1 (true), what is the output? What variable is the output dependent upon? Depends on Therefore, * 1 = 0 6. If one of the inputs is always 0 (false), what is the output? 0 Therefore, * 0 = 0 7. If = 0, then the output is 0 If = 1, then the output is 1 How does the output compare with the input? Therefore, * = 8. If = 1, then the output is 0 If = 0, then the output is 0 Therefore, * = 0 List the EIGHT basic rules of oolean lgebra you gained from exercises 1-8:

2 i. +1 =1 ii. +0= iii. += iv. *= v. * = 0 vi. *0= 0 vii. + =0 viii. *1 =1 9. How do and compare? Therefore, + = _+ How do and compare? What arithmetic law is this? Commutative Therefore, * = *

3 10. Consider the boolean expression + + C. Does it matter which OR you evaluate first : (+ or +C)? Verify that ( + ) + C = + ( + C) by writing out truth tables for each expression. Make an intermediate column for (+) and (+C) in each truth table. Here is a table to get you started: C (+) (+)+C (+C) +(+C) Now consider the boolean expression * * C. Does it matter which ND you evaluate first : (* or *C)? Verify that ( * ) * C = * ( * C) by writing out truth tables for each expression. Make intermediate columns for (*) and (*C) in each truth table. C (*) (*)*C (*C) *(*C) Which arithmetic law is this? ssociative

4 pplying boolean algebra: 12. To get into a physics program in a university, Samantha needs to have high school Physics, and either lgebra or Calculus. ssign boolean variables to the conditions and write a boolean expression for the program requirements. >> = lgebra C = Calculus P = Physics Equation: P * ( + C) 13. nother way of stating the conditions for the physics program is that Samantha needs high school Physics and lgebra, or high school Physics and Calculus. Using the same boolean variables as above, write a boolean expression for the program requirements. >> Same variables as earlier (P*) + (P*C) 14. Since both expressions refer to the same situation, the boolean expressions must be equal. Verify this by writing a truth table that includes both expressions and comparing those columns. (Model the table after the 2 input tables you used above. P C (P*) (P*)+(P*C) (P*C) John wants to go Go-Karting at KartWorld. They have conditions on who can drive their Go-Karts. You must either be over sixteen or be over twelve and have parental supervision. Using boolean variables create an expression for the Go-Karting requirements. >> P = Parental Supervision S = ge over 16 T = ge over 12 Equation: S + (P*T) 16. nother way to state the requirements, (although you wouldn t usually say it this way) is you must be over sixteen or over twelve, and, you must be over sixteen or have parental supervision. Using the same boolean variables as above, write a boolean expression for the Go-Karting requirements. >> Same variables as ealier (S+T)*(P+S)

5 17. Verify that the two expressions above are equal by writing a truth table that includes a column for each expression and comparing the results. P S T (S+T) (S+T)*(P+S) (P+S) What arithmetic law does this demonstrate for boolean algebra? Distributive 18. Given the laws you proved in problems 12-18, what can you say about (+)*(C+D)? [HINT: what algebra method can be used?] Draw a truth table to prove your hypothesis. >> (+)*(C+D) = C + D + C + D C D (*D) (*C) (*C) (*D) D + C + C + D Distributive property: Makes it so (+)*(C+D) expands to what is listed above. 19. Consider the NOT gate. Construct a truth table

6 20. Construct a truth table for the following: What conclusions can you draw from looking at the two truth tables? >> Not Not is the same as just printing it out. Not makes it the opposite

7 22. Construct truth tables for each of the following: dd more columns if you need to complete any of the tables Compare the four truth tables. What conclusions can you draw? Use boolean expression to express your findings. >> With the ands only 1 scenario evaluates to true, whereas with the or all but 1 evaluate to true These expressions are known as DeMorgan s Law.

8 24. Consider the following diagram. If =0, what is the output? 0,0. If =1, what is the output? 1,_Depends on _. Therefore, *( + ) = +*. Write a truth table for the above diagram. (+) *(+) (+) *(+) Does it look familiar? What conclusions can you draw? sends through to the and automatically at the top and the second part is 1 no matter what if is 1. If is 0 the expression will be 0 no matter what because that will make the and evaluate to false. 25. Now consider the XOR gate Create a truth table.

9 Now write a boolean expression for the following diagram ( *)+(* ) Construct a truth table. dd more columns if you need them Compare the truth tables for 25 and 26. What is your conclusion? Same thing. The more complicated version is likely for programs that aren t able to utilize the xor function. In the end the second example is just a more complex way to state the xor function

10 27. Compare the following three diagrams, by writing out the boolean expressions and then constructing corresponding truth tables. What are your conclusions? ( ), ( ), ( ) ( ) There are 2 trues in each example. In the end all three examples are the exact same thing. Within a xor it doesn t seem to matter where the not is placed, before (Example 1 and 2) or after (Example 3). ( ) ( )