1. 1 Introduction to Abstract Mathematics Applications : Digital Logic Circuits 2.4 and Number Systems 2.5 Instructor: Hayk Melikya melikyan@nccu.edu

2. 2 Introduction to Abstract Mathematics Simple electrical switching device Here are more complicated circuits

3. 3 Introduction to Abstract Mathematics Serial and Parallel switches

4. 4 Introduction to Abstract Mathematics Block Boxes and Gates An effective way to build more complicated circuits is connecting less complicated block box circuits Three such a gates: NOT-gate, AND-gate, OR-gate can be combined Black Box is specified by the signal input/output table.

5. 5 Introduction to Abstract Mathematics Combinatorial circuits 1. Never combine two input wires 2. An input line can be split and used as input for two separate gates 3. Any output can be used as input 4. No output can be feed back to gate Example: Deterring input/output table for given circuit

6. 6 Introduction to Abstract Mathematics Circuits and Boolean expressions Combinational circuit always correspond to some Boolean expression, such that input/output table of a table and a truth table of the expression are identical Construct equivalent boolean expression using disjunctive normal form as follows 1.for all outputs of 1 construct a conjunctive form based on the truth table row. 2. All conjunctive forms are united using disjunction

7. 7 Introduction to Abstract Mathematics Example: Input/output table  P  Q  R  P  Q  ~R  P  ~ Q  ~ R The circuit corresponding to given table is the disjunctions of obtained below three conjunctive terms (P  Q  R)  (P  Q  ~R)  (P  ~ Q  ~ R)

8. 8 Introduction to Abstract Mathematics Example: Here is the combinatorial circuit corresponding to the ( P  Q  R)  (P  Q  ~R)  (P  ~ Q  ~ R)

9. 9 Introduction to Abstract Mathematics Example: Construct circuit which corresponds to Exclusive or of P and Q

10. 10 Introduction to Abstract Mathematics Number Systems v Decimal number system There are only 10 digits: 0, 1, 2, 3,,4, 5, 6, 7, 8, 9 Decimal numbers are finite sequences of digits example: 376 = 3x 10 2 + 7x 10 1 + 6x10 0 = 300 + 70 + 6 v Binary number system there are only two digits: 0 and 1 Binary numbers are finite sequences of 0’s and 1’s example: 1101 = 1x2 3 + 1x 2 2 + 0x2 1 + 1x2 0 = 1x8 + 1x4 + 1x1 = 13 v Conversion between decimal and binary numbers v Binary addition and subtraction base

11. 11 Introduction to Abstract Mathematics Binary addition and subtraction Adding digits in base 2 1 + 1 = 10 2 1 + 0 = 01 2 0 + 1 = 01 2 0 + 0 = 00 2 Adding numbers in base two 1 1 1 0 1 2 + 1 0 1 0 2 1 0 0 1 1 1 2 Circuits for computer addition

12. 12 Introduction to Abstract Mathematics Digital Circuits for Addition: Full Adder – addition of two bits and a carry v Parallel Adder – addition of multi-bit numbers To construct a circuit to add multidigit binary numbers it is necessary to have circuit which computes sum of three binary digits. Such a circuit is called Full Adder

13. 13 Introduction to Abstract Mathematics Digital Circuits for Addition: Parallel Adder – addition of two 3 binary digit numbers. Two full-adders and one half adder can be used to buld a circuit to add 2 binary 3 digit numbers PQR and STU to obtain WXYZ

14. 14 Introduction to Abstract Mathematics Try: v Represent 43 in binary notation v Represent 110110 in decimal notation Add 1 1 1 0 1 0 1 + 1 0 1 1 1 1

15. 15 Introduction to Abstract Mathematics Practice problems 1. Study the Sections 1.4 and 1.5 from your textbook. 2. Be sure that you understand all the examples discussed in class and in textbook. 3. Do the following problems from the textbook: Exercise 2.4, # 2, 4, 15, 19, 23. Exercise 2.5, # 3, 5, 8, 10, 14, 18.