Computer Science 210 Computer Organization Introduction to Boolean Algebra

George Boole English mathematician (1815-1864) Boolean algebra Logic Set Theory Digital circuits Programming: Conditions in while and if statements

Boolean Constants In Boolean algebra, there are only two constants, true and false Boolean constant Binary digit State of a switch Voltage level true 1 On +5V false Off 0V

Boolean Variables Boolean variables are variables that store values that are Boolean constants. Let A be true Let B be false Etc.

Boolean Operator AND If A and B are Boolean variables (or expressions) then A AND B is true if and only if both A and B are true.

Boolean Operator AND If A and B are Boolean variables (or expressions) then A AND B is false if and only if either A or B are false or they’re both false.

Boolean Operator AND We denote the AND operation like multiplication in ordinary algebra: AB or A.B

Boolean Operator OR If A and B are Boolean variables (or expressions) then A OR B is true if and only if at least one of A and B is true.

Boolean Operator OR If A and B are Boolean variables (or expressions) then A OR B is false if and only if both A and B are false.

Boolean Operator OR We denote the OR operation like addition in ordinary algebra: A+B

Boolean Operator NOT If A is a Boolean variable (or expression) then NOT A has the opposite value from A.

Boolean Operator NOT We denote the NOT operation by putting a bar over the variable (or expression) _ A Or use tilde (~) when bar is not available: ~A

Boolean Expressions As with ordinary algebra, a Boolean expression is a well-formed expression made from Boolean constants Boolean variables Operators AND, OR and NOT Parentheses Example: __ ____ AB + (A+C)B

Evaluating a Boolean expression At any time, the value of a BE can be computed using the current values of the variables. __ AB + (CD) Let A = true Let B = false Let C = true Let D = false Then the resulting value is true

Operator precedence NOT comes first, then AND, and finally OR (Like arithmetic negation, product, and addition) A + BC is not the same as (A + B)C

Evaluating a Boolean expression Unlike ordinary algebra, for a BE, there are only finitely many possible assignments of values to the variables; so, theoretically, we can make a table, called a truth table, that shows the value of the BE for every possible set of values of the variables. For convenience, use 0 = false 1 = true

Truth Table for AND A B AB 1

Truth Table for OR A B A+B 1

Truth Table for NOT A _ 1

Filling in a Truth Table If there are N variables, there are 2N possible combinations of values Thus, there are 2N rows in the truth table Fill in the values by counting up from 0 in binary

Example Construct a truth table for _ ___ E = AB + (A+C)B

_ ___ E = AB + (A+C)B A B C 1 Assign the values of the variables first

_ ___ E = AB + (A+C)B Then add columns for each operation A B C 1 _ B 1 _ B 1 Then add columns for each operation

_ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1

_ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1 A+C 1

_ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1 A+C 1 ___ (A+C) 1

_ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1 A+C 1 ___ (A+C) 1 ___ 1 _ B 1 _ AB 1 A+C 1 ___ (A+C) 1 ___ (A+C)B 1

_ ___ E = AB + (A+C)B A B C 1 _ B 1 _ AB 1 A+C 1 ___ (A+C) 1 ___ 1 _ B 1 _ AB 1 A+C 1 ___ (A+C) 1 ___ (A+C)B 1 E 1