BOOLEAN ALGEBRA SUMMARY We can interpret high or low voltage as representing true or false. A variable whose value can be either 1 or 0 is called a Boolean variable. AND, OR, and NOT are the basic Boolean operations. We can express Boolean functions with either an expression or a truth table. Every Boolean expression can be converted to a circuit. Now, we’ll look at how Boolean algebra can help simplify expressions, which in turn will lead to simpler circuits. BOOLEAN OPERATOR PRECEDENCE The order of evaluation is: 1. Parentheses 2. NOT 3. AND 4. OR Examples x (y' + z) xy+z BOOLEAN F UNCTIONS A Boolean function Binary variables Binary operators OR and AND Unary operator NOT Parentheses Examples BOOLEAN ALGEBRA POSTULATES Commutative Law x•y=y•x Identity Element x1=x x’ 1 = x’ Complement x • x’ = 0 x+y=y+x x+0=x x’+ 0 = x’ x + x’ = 1 BOOLEAN ALGEBRA THEOREMS Theorem 1 Theorem 2 Theorem 3:Involution BOOLEAN ALGEBRA THEOREMS Theorem 4:Associative: Theorem 4: Distributive Theorem 5: DeMorgan Theorem 6: Absorption TRUTH TABLE TO VERIFYDEMORGAN’S PROOFS: PROOFS AND EXAMPLES: CONVERTING FROM TRUTH TABLE TO BOOLEAN FUNCTION This problem will be solved by showing that any Boolean function can be represented by a Boolean sum of Boolean products of the variables and their complements or the product of sums. There are two ways to convert from truth tables to Boolean functions: 1. Using Sum of Products /Minterms 2. Using Product of Sums /Maxterms CONVERTING FROM TRUTH TABLE TO BOOLEAN FUNCTION Minterm – Product (AND function) – Contains all variables – Evaluates to ‘1’ for a specific combination CONVERTING FROM TRUTH TABLE TO BOOLEAN FUNCTION Maxterm – Sum (OR function) – Contains all variables – Evaluates to ‘0’ for a specific combination CONVERTING FROM TRUTH TABLE TO BOOLEAN FUNCTION CONVERTING FROM TRUTH TABLE TO BOOLEAN FUNCTION Sum of Minterms CONVERTING FROM TRUTH TABLE TO BOOLEAN FUNCTION Product of Max Term CONVERT SOP TO STANDARD SOP FORM Step 1: Find the missing literal in each product term if any. Step 2: And each product term having missing literals with terms formed by ORing the literal and its complement. Step 3: Expends the term by applying, distributive law and reorder the literals. Step 4: Reduce the repeated product terms. Because A + A = A (Theorem 1a ). CONVERT SOP TO STANDARD SOP FORM CONVERT SOP TO STANDARD SOP FORM CONVERT POS TO STANDARD POS FORM Step 1: Find the missing literal in each sum term if any. Step 2: OR each sum term having missing literals with terms form by ANDing the literal and its complement. Step 3: Expends the term by applying, distributive law and reorder the literals. Step 4: Reduce the repeated product terms. Because A + A = A (Theorem 1a ). CONVERT POS TO STANDARD POS FORM CONVERT POS TO STANDARD POS FORM