Q2.Simplfy boolean expression using boolean algebra(bc'+a'd)(ab+cd')Boolean algebra is the category of algebra in which the variable’s values are the truth values, true and false, ordinarily denoted 1 and 0 respectively. It is used to analyze and simplify digital circuits or digital gates. It is also called Binary Algebra or logical Algebra Boolean Expression A logical statement that results in a Boolean value, either be True or False, is a Boolean expression. Sometimes, synonyms are used to express the statement such as ‘Yes’ for ‘True’ and ‘No’ for ‘False’. Also, 1 and 0 are used for digital circuits for True and False, respectively. Laws of Boolean Algebra There are six types of Boolean algebra laws. They are: Commutative law Associative law Distributive law AND law OR law Inversion law Those six laws are explained in detail here. Commutative Law Any binary operation which satisfies the following expression is referred to as a commutative operation. Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit. A. B = B. A A + B = B + A Associative Law It states that the order in which the logic operations are performed is irrelevant as their effect is the same. ( A. B ). C = A . ( B . C ) ( A + B ) + C = A + ( B + C) Distributive Law Distributive law states the following conditions: A. ( B + C) = (A. B) + (A. C) A + (B. C) = (A + B) . ( A + C) AND Law These laws use the AND operation. Therefore they are called AND laws. A .0 = 0(NULL Law) A . 1 = A A. A = A(Idempotant law) OR Law These laws use the OR operation. Therefore they are called OR laws. A + 0 = A A + 1 = 1 A + A = A Inversion Law In Boolean algebra, the inversion law states that double inversion of variable results in the original variable itself. A''=A Boolean Algebra Theorems The two important theorems which are extremely used in Boolean algebra are De Morgan’s First law and De Morgan’s second law. These two theorems are used to change the Boolean expression. This theorem basically helps to reduce the given Boolean expression in the simplified form. These two De Morgan’s laws are used to change the expression from one form to another form. Now, let us discuss these two theorems in detail. De Morgan’s First Law: De Morgan’s First Law states that (A.B)’ = A’+B’. The first law states that the complement of the product of the variables is equal to the sum of their individual complements of a variable.
Q3.What is demorgan theorem(bc'+a'd)(ab+cd') abc'+bc'cd'+a'dab'+a'dcd' now lets remove obsolete terms cc'=0(According to boolean algebra identity law). abc'+0+0+0 =abc'(Using boolean algebra simplfy expression result)
Q4.Using demorgan theorem show that (A+B)'(A'+B')'=0(A+B)'=A'.B' Demorgan theorem state that the complement of a sum equals the product. (A.B)'=A'+B' It states that the complement of product equals the sum. Let prove A B A' B' A+B (A+B)' A'.B' 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 Vales of (A+B)'=A'.B'(According to the demorgan law).
Q5 simplfy using boolean alegbra (A+B)(A+C)(A+B)'(A'+B')'=0 According to demorgan law (A+B)'=A'.B' so replace (A+B)' with A'.B' (A'.B')(A.B) A'A=0 According to the identity law. A'ABB' 0 Proved (A+B)'(A'+B')=0
(A + B)(A + C) Again, multiplying out the brackets gives us: (A + B)(A + C) Start multiply: A.A + A.C + A.B + B.C Distributive law but: A.A = A Idempotent Law then: A + A.C + A.B + B.C Reduction however: A + A.C = A Absorption Law thus: A + A.B + B.C Distributive Law again: A + A.B = A Absorption Law thus: A + B.C Result Then the Boolean expression of (A + B)(A + C) can be reduced to just “A + B.C” using the various Boolean algebra laws.
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