1 introduction to abstract mathematics applications : digital logic circuits 2.4 and number systems...

1Introduction to Abstract Mathematics

Applications : Digital Logic Circuits 2.4 and Number Systems 2.5

Instructor: Hayk Melikya [email protected]


Simple electrical switching device

Here are more complicated circuits


Serial and Parallel switches


Block Boxes and Gates

An effective way to build more complicated circuits is connecting less complicated block box circuitsThree such a gates: NOT-gate, AND-gate, OR-gate can be combined

Black Box is specified by the signal input/output table.


Combinatorial circuits

1. Never combine two input wires2. An input line can be split and used as input for two

separate gates3. Any output can be used as input4. No output can be feed back to gate

Example: Deterring input/output table for given circuit


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


Example:Input/output table

P Q R

P Q ~R

P ~ Q ~ R

The circuit corresponding to given tableis the disjunctions of obtained below three conjunctive terms

(P Q R) (P Q ~R) (P ~ Q ~ R)


Example:Here is the combinatorial circuit corresponding to the

( P Q R) (P Q ~R) (P ~ Q ~ R)


Example: Construct circuit which corresponds to Exclusive or of P and Q


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 = 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

Conversion between decimal and binary numbers Binary addition and subtraction

base

base


Binary addition and subtraction Adding digits in base 2 1 + 1 = 102

1 + 0 = 012

0 + 1 = 012

0 + 0 = 002

Adding numbers in base two 1 1 1 0 12

+ 1 0 1 02

1 0 0 1 1 12 Circuits for computer addition


Digital Circuits for Addition: Full Adder – addition of two bits and a carry

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


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


Try: Represent 43 in binary notation Represent 110110 in decimal notation

Add 1 1 1 0 1 0 1+ 1 0 1 1 1 1


Practice problems1. 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.

1 introduction to abstract mathematics applications : digital logic circuits 2.4 and number systems...

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