# VERY URGENT DISCRET MATH

CPST-1070 Mathematics for Information Technology

Spring 2020 Test 2

Name____________________________________________

True or False (2 point each)

____ 1. The following is a proposition: Today it is raining.

____ 2. AB’C v ABC’ v A’B’C’ is in disjunctive normal form.

____ 3. (pq)’ ≡ p’ v q’

____ 4. The following is the correct drawing of an OR gate:

____ 5. A Karnaugh map can be used to simplify any Boolean expression.

____ 6. Any Boolean expression can be represented by a logic circuit.

Problems (6 points each)

Show your work if you want partial credit!

7. Let W represent the statement “The Baby Cakes Win

Let B represent the statement “I like the name Baby Cakes”

Translate the following logical symbols into English: W v ~ B

8. Write the negation of the statement: Math is hard or I am stupid

9. Give the truth table for: AB V A’B’

10. Given the Boolean function f(A,B,C) = A’BC v A’BC v AB’C’, what is the value of this function when A=1, B=0, and C=0, i.e, what is f(1,1,0)?

11. Fill in the following Boolean function table for the expression:

f(A,B,C) = ABC v ABC’ v A’B’C v A’BC’ v AB’C’

 A B C f(A,B,C)

12. Write the Boolean expression associated with the following logic gate diagram:

13. Below is the logic gate circuit for the XOR (Exclusive OR) function which produces an output of 1 when exactly one of a and b are 1:

Write a Boolean expression associated with this logic gate circuit.

14. Draw a logic gate diagram for the expression: (A’vB’)C

15. A “nand” gate is a special logic gate circuit for the nand logic function. The nand function if defined as follows: A nand B = (AB)’. It has as its truth table:

 A B A nand B 1 1 0 1 0 1 0 1 1 0 0 1

Draw a logic gate circuit (using AND, OR and/or NOT gates) for the nand function.

16. Define the Boolean function f as follows:

 X Y Z f(X,Y,Z) 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1

Write the Boolean expression in disjunctive normal form for this function.

17. Given the Boolean expression:

ABC v ABC’ v A’B’C v A’BC v A’BC’

Draw the Karnaugh Map for this expression and circle a minimal number of groups of 1’s that covers all the 1’s.

18. The following is the Karnaugh map for the expression:

ABC v ABC’ v AB’C v AB’C’ v A’B’C’ v A’BC’

 AB AB’ A’B’ A’B C 1 1 C’ 1 1 1 1

Use this Karnaugh map to simplify the expression.

ABC v ABC’ v AB’C v AB’C’ v A’B’C’ v A’BC’ ≡ _______________________________

19. The following is the Karnaugh map for the expression:

ABCD v ABCD’ v AB’CD’ v A’B’CD’ v A’B’C’D’ v A’BCD v A’BCD’ v A’BC’D’ v A’BC’D

 AB AB’ A’B’ A’B CD 1 1 CD’ 1 1 1 1 C’D’ 1 1 C’D 1

Use this Karnaugh map to simplify the expression.

Application Problem (10 Points)

20. A locking mechanism for a safety deposit box at a bank consists of 2 keys and a thumbprint scanner. One key is kept by the bank, and the 2nd key is given to the owner of the safety deposit box. The thumbprint scanner recognizes the thumbprint of the owner of the safety deposit box. The safety deposit box opens under the following conditions:

The bank key is inserted and either the owners key is inserted or the owner’s thumbprint is recognized (or both)

Or

The owner’s key is inserted and the owner’s thumbprint is recognized.

You need to create a control unit for this locking mechanism. To do so, you will need to create a logic gate circuit that that takes as inputs whether or not the bank’s key or the owner’s key are inserted and whether or not the owner’s thumbprint is recognized, and as an output whether or not the lock should be opened or not.

Let A be the bank’s key is inserted

Let B be the owner’s key is inserted

Let C be the thumb scanner recognizes the owners thumbprint

a) Fill in the following truth table that reflects the appropriate input and output values

 A B C Box opens 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0

b) From this truth table, write a Boolean expression in disjunctive normal form

c) Complete the Karnaugh Map for the expression in part b

 AB AB’ A’B’ A’B C C’

d) Use the Karnaugh Map to reduce the expression to a simpler form

e) Draw a gated logic circuit using the simple expression from part d.

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