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Applications : Digital Logic Circuits 2.4 and Number Systems 2.5 Instructor: Hayk Melikya Introduction to Abstract Mathematics [email protected] 1 Simple electrical switching device Here are more complicated circuits Introduction to Abstract Mathematics 2 Serial and Parallel switches Introduction to Abstract Mathematics 3 Block Boxes and Gates Black Box is specified by the signal input/output table. 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 Introduction to Abstract Mathematics 4 Combinatorial circuits 1. 2. 3. 4. Never combine two input wires An input line can be split and used as input for two separate gates Any output can be used as input No output can be feed back to gate Example: Deterring input/output table for given circuit Introduction to Abstract Mathematics 5 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 Introduction to Abstract Mathematics 6 Example: Input/output table The circuit corresponding to given table is the disjunctions of obtained below three conjunctive terms P Q R P Q ~R P ~ Q ~ R (P Q R) (P Q ~R) (P ~ Q ~ R) Introduction to Abstract Mathematics 7 Example: Here is the combinatorial circuit corresponding to the ( P Q R) (P Q ~R) (P ~ Q ~ R) Introduction to Abstract Mathematics 8 Example: Construct circuit which corresponds to Exclusive or of P and Q Introduction to Abstract Mathematics 9 Number Systems 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 102 + 7x 101 + 6x100 base = 300 + 70 + 6 Binary number system there are only two digits: 0 and 1 Binary numbers are finite sequences of 0’s and 1’s example: 1101 = 1x23 + 1x 22 + 0x21 + 1x20 = 1x8 + 1x4 + 1x1 = 13 base Conversion between decimal and binary numbers Binary addition and subtraction Introduction to Abstract Mathematics 10 Binary addition and subtraction Adding numbers in base two 1 1 1 0 12 + 1 0 1 02 1 0 0 1 1 12 Circuits for computer addition Introduction to Abstract Mathematics Adding digits in base 2 1 + 1 = 102 1 + 0 = 012 0 + 1 = 012 0 + 0 = 002 11 Digital Circuits for Addition: Full Adder – addition of two bits and a carry 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 Parallel Adder – addition of multi-bit numbers Introduction to Abstract Mathematics 12 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 Introduction to Abstract Mathematics 13 Try: Represent 43 in binary notation Represent 110110 in decimal notation Add 11101 01 + 1011 11 Introduction to Abstract Mathematics 14 Practice problems 1. 2. 3. Study the Sections 1.4 and 1.5 from your textbook. Be sure that you understand all the examples discussed in class and in textbook. Do the following problems from the textbook: Exercise 2.4, # 2, 4, 15, 19, 23. Exercise 2.5, # 3, 5, 8, 10, 14, 18. Introduction to Abstract Mathematics 15