Boolean Algebra: Overview, Questions, Preparation

Boolean algebra is a part of algebra that deals with the connection between entities and comes under mathematical logic associated with two values—0 and 1—where one denotes true, and 0 represents false.

Terminologies Related to Boolean Algebra

Boolean variable:- Boolean variables are the type of variables that only deal with two outcome values: true(1) or false(0).

Literals:- Literals are used in the representation of the numerical value. They are used as a variable or its complement in any expression.

Boolean function:- A Boolean function is a type of function that deals with several variables, say “n”. The total combinations of this variable n will be “2n”. Boolean functions operate with only 0 & 1 as its outputs.

Complement:- Boolean complements evaluate the opposite of the original input. It is represented by a bar.“_”

Truth table:- The truth table is a table used to represent a Boolean expression with every possible combination of inputs in a tabular manner. 

Boolean Operators

The three main Boolean operators are:
NOT (~)
AND (^)
OR (v)

Negation of a Statement (NOT)

The negative of any statement is its negation. Its symbol is ~.

Its truth table is:

Notice that the NOT operator turns true into false and false into true.

Conjunction (AND) 

This operator combines two given statements using AND. its symbol is ^.

Its truth table is:

Notice that it returns true only if both the statements are true.

Disjunction (OR) 

This operator combines two given statements using OR. its symbol is v.

Its truth table is:

Notice that it returns true even if one of the statements is true.

There are six types of Boolean laws:

Commutative law:- When the order of operands of a boolean expression is interchanged, the results remain unaffected. Hence, according to this law interchanging the values will not cause any alteration in the result.

For example, A + B = B + A
A * B = B * A

Associative law:- If A, B and C are three variables, then using the OR operator on the variable yields the same result even if braces swap the grouping of variables. 
example, (A+B)+C=A+(B+C)

A * (B * C) = (A * B) * C

Distributive law:- If A, B and C are three variables, then using the AND operator or OR operator on the variables will not produce any change even if the multiplication result is added with a variable.

example, A + BC = (A + C)(A + B).

A(B + C) = (AB) + (AC)

Absorption law:- Absorption Law deals with linking of any two binary operations. It is also known as absorption identity.
example, A + (A * B) = A

A * (A + B) = A

De Morgan’s Theorem

De Morgan’s theorem or law is the concept of interchanging variables and operators within a function like replacing 1 with 0 and 0 with 1, OR operator with AND operator and vice versa. This concept is also known as Duality. The complement either of the product or sum of two variables is equated to the individual complements’ sum or product for the respective variables.

(A * B)’ = A’ + B’

(A + B)’ = A’ * B’

This chapter carries 3-5 marks in your class 11th exams. Prepare it nicely as it is one of the easiest one!

1. Draw a logic circuit for the following Boolean expression: ab + c.d'.

Solution.

Source : NCERT

2. Verify the following using Boolean Laws :

LT + V = LTV + LP.V + U.V

Solution.
L.H.S.
= U’ + V
= U’ . (V + V) + V (LP + U)
= U’ . V + LP . V + U . V + U. V
= U’. V + LP. V + U. V
= R.H.S.
OR
R.H.S.
= U’V’ + U’. V + U. V
= LP . (V + V) + U. V
= U’ 1 + U.V
= U’ + U.V
= U’ +V
= L.H.S.

3. Verify the following using Boolean Laws X + Z = X + X’. Z + Y. Z

Solution.

Taking RHS
X + X’Z + YZ
= (X + X’). (X + Z) + YZ (Distribution Law)
= 1. (X + Z) + YZ    (A + A’ = 1)
= X + Z + YZ
= X + Z (1 + Y)
= X + Z    (1 + A = 1; 1. A = A)
= Hence verified

Q: What are the applications of Boolean algebra? 

Q: What are the three major Boolean operators?

Q: Who formulated Boolean algebra?

Q: Why is Boolean algebra called “Boolean” algebra?

Q: Can any random number be used instead of “0” or “1” in Boolean operations?