Question

Reduce the expression f=ab'C+B+BD'+abd +a'c​

Answer 1

Explanation:

BOOLEAN THEOREMS:

Boolean algebra helps to analyze a logic circuit and express its operation mathematically. We have several Boolean Theorems that helps us to simplify logic expressions and logic circuits.

Single Variable Theorems:

AND Operation OR Operation

A . 0 = 0 A + 0 = A

A . 1 = A A + 1 = 1

A . A = A A + A = A

A . A’ = 0 A + A’ = 1

Multivariable Theorems:

The multivariable theorems involves more than one variable.

Commutative laws:

A + B = B + A

A . B = B . A

Associative laws:

A + (B + C) = (A + B) + C = A + B + C

A (BC) = (AB) C = ABC

Distributive laws:

A (B + C) = AB + AC

A + BC = (A+B) (A+C)

(A + B) (C + D) = AC + AD + BC + BD

Absorption laws:

A + AB = A

A (A + B) = A

Redundant Literal Rule:

A + A’B = A + B

A’ + AB = A’ + B

A(A’ + B) = AB

Demorgan’s Theorem:

(A+B)’ = A’ . B’

(AB)’ = A’ + B’

This law can be extended to any number of variables or combinations of variables.

How to apply Demorgan’s theorem to an expression?

Complement the entire given expression.

Change all the AND’s to OR’s and all the OR’s to AND’s.

Complement each of the individual variables.

Change all 0s to 1s and all 1s to 0s.

This procedure is called as Demorganization or Complementation of switching expressions. In simple words, we can say it as ‘Break the line, change the sign’.

f(A, B, ………, 0, 1, . , +) = f(A’, B’, ………., 1, 0, +, .)

Consensus Theorem:

AB + A’C + BC = AB + A’C

(A + B)(A’ + C)(B + C) = (A + B)(A’ + C)

If one term is containing A and the other term is containing A’ and the third term containing the left-out literals of the first two terms then the third term is redundant. It means the function remains same with and without the third term.

This theorem can be extended to any number of variables.

AB + A’C + BCD = AB + A’C

(A + B)(A’ + C)(B + C + D) = (A + B)(A’ + C)

Transposition Theorem:

AB + A’C = (A + C)(A’ + B)

DUAL OF A BOOLEAN FUNCTION:

The DUAL of a boolean function is obtained by interchanging OR and AND operations and replacing 1’s by 0’s and 0’s by 1’s.

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