Boolean logic

introduction to boolean logic,truth table and types of forms in truth table , evaluate the expression using truth table
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1. BOOLEAN LOGIC CLASS: XI COMPUTER SCIENCE -083

2. INTRODUCTION TO BOOLEAN LOGIC Boolean Logic is a form of algebra which is centered around three simple words known as Boolean Operators: “Or,” “And,” and “Not”. At the heart of Boolean Logic is the idea that all values are either true or false MEANS 1’s [TRUE] or 0’s[FALSE]. Boolean logic was originally developed by George Boole in mid-1800s.Boolean logic is a very easy way to figure out the truth of a expression using True or False. 10 > 5 TRUE

3. Boolean logic is very easy to explain and to understand. The values that expressed only in the form of 1’s means TRUE/YES or 0’s means FALSE/NO. Consider the example: Should I go to restaurant or not? Should I purchase a new mobile or not? All these statements have only one answer, either in the form of yes/no, True/False.

4. Boolean Variable. It is also known as binary variable or logical variable that takes values from Boolean algebra. A Boolean variable can take only one binary – valued quantity out of two possible values i.e., YES/TRUE/1 or NO/FALSE/0. One bit represents one Boolean variable. So in electrical circuit, if the signals passes, it represents 1, else represented through 0.

5. Boolean Constant: The values which are stored in binary variables are known as Boolean Constants. Therefore, the values are YES/TRUE/1’s or NO/FALSE/0’s. Boolean Operators: And Or Not (Dot) + (Plus) ( ) or ( ‘ ) ^ (caret) ( ~ ) स्थिर ,stable , static, stationary, constant, stagnant, still ( )^ Reversed caret

6. And (Dot) AND operator is a binary operator that operates on two variable and the operation performed by AND operator is known as logical multiplication. ( ^ ) Consider the following two variable values and find the result using AND operator: A B A . B 1 1 1 1 1 A B A ^ B F F T T F T F T F F F T A B ^

7. OR OR operator is a binary operator that operates on two variable and the operation performed by OR operator is known as logical addition. Consider the following two variable values and find the result using OR operator: A B A + B 1 1 1 1 1 1 1 A B A + B F F T T F T F T F T T T + (Plus) ^ A B ^

8. NOT NOT operator is a unary operator that operates on one variable and the operation performed by NOT operator is known as Negation or complementation.. Consider the following one variable values and find the result using NOT operator: A 1 1 A A’ F T T F ~ A ( ) or ( ‘ ) , ( ~ ) A’ A TRUE NOT = FALSE

9. TRUTH TABLE: A truth table is a mathematical table used to determine if a compound statement is true or false. In a truth table , each statement is typically represented by a letter or variable, like p, q, or r, and each statement also has its own corresponding column in the truth table that lists all of the possible truth values Three types of truth table form: Two variable form Three variable form Four variable form

10. How to Create a Truth Table The first step in creating a truth table is to determine the number of variables. Draw COLUMNS for each unique variable. For example if two variable A,B then two columns Draw ROWS which can be calculated as 2 n where n is total number of variables. Write 0’s in the first half for the number of rows in the first column and 1’s in the next half in the first column Similarly, Write 0’s in the first half for number of rows which have 0’s in the first column and 1’s in the same number. Repeat this pattern till it written for all.

11. Let us discuss First two variable form A,B : It means two variable so number of rows are 2 2= 4 A B Write 0’s in the first half for the number of rows in the first column and 1’s in the next half in the first column 1 1 Draw the 2 column for A, B variable Similarly, Write 0’s in the first half for number of rows which have 0’s in the first column and 1’s in the same number. Repeat this pattern till it written for all. 1 1 Method 1: LEFT To RIGHT

12. Let us discuss First two variable form A,B : It means two variable so number of rows are 2 2= 4 A B 1 1 Draw the 2 column for A, B variable 1 1 Start from the last column and put 0,1,0,1…… upto the last row Now go to next column at the left and now again write 0,1 but double it like : 0,0,1,1,0,0,1,1……… Method 2: RIGHT To LEFT

13. It means two variable so number of rows are 2 3 =8 rows A B C Draw the 3 column for A, B, C variable A B C First column half 0’s and half 1’s 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Three variable form A,B,C. LEFT To RIGHT RIGHT To LEFT Method 1: Method 2: Write 0’s in the first half for number of rows which have 0’s in the first column and 1’s in the same number. Repeat same for next Start from the last column and put 0,1,0,1 next column at the left and now again write 0,1 but double it like : 0,0,1,1,….

14. Three variable so number of rows are 2 4 =16 rows A B C D 1 1 1 1 Four variable form A,B,C,D. LEFT To RIGHT RIGHT To LEFT Method 1: Method 2: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 First column half 0’s and half 1’s Write 0’s in the first half for number of rows which have 0’s in the first column and 1’s in the same number. Repeat same for next A B C D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 next column at the left and now again write 0,1 but double it like : 0,0,1,1,…. Start from the last column and put 0,1,0,1

15. Now you know how to draw truth table Let us solve the Boolean expression using truth table Rules to evaluating Boolean expression: Evaluate the Boolean expression from left to right Evaluate the expression in parenthesis first Perform all NOT expression Perform all AND expression Perform all OR expression

16. Evaluate the Boolean expression A+BC’ using truth table. How many variable 3 A,B,C mean 2 3 =8 rows. A B C 1 1 1 1 1 1 1 1 1 1 1 1 C’ BC’ A + BC’ 1 1 1 1 1 1 1 1 1 1 1 1.Perform all NOT expression 2.Perform all AND expression 3.Perform all OR expression BC’ means B.C’

17. Evaluate the Boolean expression A+ABC’+A’C’ using truth table. How many variable 3 A,B,C mean 2 3 =8 rows. A B C 1 1 1 1 1 1 1 1 1 1 1 1 A’ C’ ABC’ A’C’ A+ABC’+A’C’ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

18. Verify using truth table that ( X + Y ) ‘ = X’ Y’ for each X,Y{0,1}. X Y 1 1 1 1 X + Y 1 1 1 (X + Y)’ 1 X’.Y’ 1 X’ 1 1 Y’ 1 1 (LHS) (RHS) It is proved that LHS = RHS