01 March 2009Instructor: Tasneem Darwish1 University of Palestine Faculty of Applied Engineering and Urban Planning Software Engineering Department Introduction to Discrete Mathematics Introduction

01 March 2009Instructor: Tasneem Darwish2 Outlines Number systems. The importance of binary number. binary arithmetic. Boolean algebra. Basic Boolean operations. Boolean expressions. Basic Boolean theorems.

01 March 2009Instructor: Tasneem Darwish3 Number systems The decimal number system is a base 10 system. 0,1, 2, 3, 4, 5, 6, 7, 8, 9 The binary number system is a base two system. 0, 1

01 March 2009Instructor: Tasneem Darwish4 The importance of binary numbers Computers are based on the logic of binary numbers. Any digital system is designed according to the binary systems notion. In computers every character is saved in memory as 8bits. Each bit is represented as either 0 or 1. Example: The character ‘A’ is saved as

01 March 2009Instructor: Tasneem Darwish5 Binary Arithmetic Arithmetic operations in digital systems are usually done in binary because design of digital components to perform binary arithmetic is much easier than for decimal. The binary arithmetic is much easier than decimal. In binary number system there is four main operations: Addition. Subtraction. Multiplication. Division.

01 March 2009Instructor: Tasneem Darwish6 Binary Arithmetic Binary addition has four cases: 0+0=0 0+1=1 1+0=1 1+1= 1 and a carry 1 Binary multiplication also has four rules: 0*0=0 1*0=0 0*1=0 1*1=1

01 March 2009Instructor: Tasneem Darwish7 Boolean Algebra Boolean variables, such as X or Y, are used to represent the input or output of a digital electronics components. Boolean variables can take only two values ‘0’ or ‘1’ The ‘0’ value is to indicate low voltage. The ‘1’ value is to indicate high voltage. Although the ‘0’ and ‘1’ symbols look like binary numbers, but they are not because they have no numeric value

01 March 2009Instructor: Tasneem Darwish8 Basic Boolean Operations There are three basic Boolean operations:  And  Or  Complement (inverse) Boolean operations are usually defined using truth table. A truth table describes all the possible values of the operation arguments and their results.

01 March 2009Instructor: Tasneem Darwish9 Basic Boolean Operations The And operation Example: A. B = X (pronounced: A and B equals X) The truth table for the And operation is as follows: ABA.B=X

01 March 2009Instructor: Tasneem Darwish10 Basic Boolean Operations The OR operation Example: A + B = X (pronounced: A or B equals X) The truth table for the OR operation is as follows: ABA+B=X

01 March 2009Instructor: Tasneem Darwish11 Basic Boolean Operations The Complement operation Example: A` = X (pronounced: A inverse equals X) The truth table for the Complement operation is as follows: AA`=X 01 10

01 March 2009Instructor: Tasneem Darwish12 Boolean Expressions Boolean expressions consist of Boolean variables with one or more Boolean operations. Example: X= (A.B) +C The truth table for this expression is as follows: ABCA.BX=(A.B)+C

01 March 2009Instructor: Tasneem Darwish13 Basic Boolean Theorems These are the basic Boolean algebra theorems:  X+0=XX.0=0  X+1=1X.1=X  X+X=XX.X=X  (X`)`=X  X+X`=1X.X`=0