Logic gate
The basic building blocks of digital circuitry are logic gates. They are called so because they manipulate the binary data in the logical way. Logical gates can be constructed from diodes, transistors and registers. These are connected in such a way that the circuit output is the result of logic operation (AND, OR, NOT etc performed on the inputs.
Truth table
A logic circuit depends on the logic level present at the circuit’s input. A simple way to show the output state for each possible input state combination is by a truth table. Note that there are 4 input combination for the two-input truth table, 8 combination for the three-input truth table, 16 combination for the four-input truth table. The number of input combinations will equal 2N for N-input truth table.
Inverter & Boolean expression
NOT gate is commonly called an INVERTER. The circuit always has only a single input and its output logic level is always opposite to the input logic level.
Commutative laws, associative laws and distributive lws
Theorem 1: A+B = B+A
Theorem 1: A.B = B.A
Theorem 1 and 2 are called the commutative laws. These laws indicate that the inputs of a gate are equivalent and interchangeable.
Theorem 3: (A.B).C = A.(B.C)
Theorem 4: (A+B)+C = A+(B+C)
Theorem 3 and 4 are called the associative laws which sates that the order in which variables AND or OR does not matter.
Theorem 5 : A.(B+C) = AB + AC
Theorem 6 : (A+B).(C+D) = AC + BC + AD + CD
Theorem 5 and 6 are called the distributive laws which states that an expression can be expanded by multiplying term by term just the same as in ordinary algebra. These theorems also indicate we can also factor out an expression the common variable line in ordinary algebra.
DeMorgan’s theorems.
DeMorgan’s theorems are extremely useful in simplifying expressions in which a product or sum of variables is inverted. The two theorems are :
Theorem 1 says that when OR sum of two variable is inverted, this is the same as inverting each variable individually and then ANDing this inverted variables.
Theorem 2 says that when the AND product of variables is inverted, this is the same as inverting each variable individually and then ORing them.
Sum-of-products and product-of-sums form of the logic expression
Each of these sum-of-products expressions consisits of two or more And terms (products) that are ORed together. Each AND term consists of one or more variables in either complemented or uncomplemented form. In a sum-of-products expression, one inversion sign can not cover 
more than one variable in a term (e.g. we can not have ABC or RST).
more than one variable in a term (e.g. we can not have ABC or RST).
Product-of-sums form:
These is another general form of logic expressions that is sometimes used in logic circuit design. It is called the product-of-sums form and it consists of two or more OR term contains one or more variables in complemented or uncomplemented form.
Types of simplification.
There are two types of simplification:
i) Simplification using Boolean algebra.
ii) Karnaugh mapping method.
d)Write down the steps of algebraic simpoication.
The algebraic simplification contains two essential steps:
i) The original expression is put into the sum-of-products form by repeated application of DeMorgan’s theorems and multiplication of terms.
ii) Once it is in the form, the products terms are checked for common factor factorizing is performed wherever possible.
The steps for simplifying a Boolean expression using karnaugh map
The steps below are followed in using the K-map method for simplifying a Boolean expression.
i) Construct the K-map and place 1’s in those squares corresponding to the 1’s in the truth table. Place 0’s in the other square.
ii) Examine the map for adjacent 1’s and loop those 1’s which are not adjacent to any other 1’s. These are called isolated 1’s.
iii) Next, look at those 1’s which are adjacent to only one other 1. loop any pair containing such a 1.
iv) Loop any octate even it contains 1’s that have already been looped.
v) Loop any quad contains one or more 1’s which have not already been looped, making sure to use the minimum numbers of loops.
vi) Loop any pair necessary to include any 1’s that have not yet been looped, making sure to use the minimum number of jobs.
vii) Form the OR sum of all terms generated by each loop.
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