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