Assoc. Prof. Dr. Ahmet Turan ÖZCERİT

 Boolean postulate  Simplifying boolean equations  Truth tables You will learn: 2

He/She can describe boolean algebra postulate 3  Logic gates and logic circuits are designed by the help of boolean algebra  Boolean Algebra was invented by George Boole in 1850s  C.E. Shannon adapted boolean algebra to switching circuits and finalized it for modern digital designs in 1930s  Boolean algebra uses three fundamental operations: AND, OR and NOT  All boolean equations and all logic units can be defined by these three operators  Arithmetic operations can also be implemented by AND, OR and NOT  Logic gates and logic circuits are designed by the help of boolean algebra  Boolean Algebra was invented by George Boole in 1850s  C.E. Shannon adapted boolean algebra to switching circuits and finalized it for modern digital designs in 1930s  Boolean algebra uses three fundamental operations: AND, OR and NOT  All boolean equations and all logic units can be defined by these three operators  Arithmetic operations can also be implemented by AND, OR and NOT

He/She can describe boolean algebra postulate 4 1A + 0 = AA. 1 = Aidentity 2A + A’ = 1A. A’ =0complement 3A + B = B+ AA. B = B. Acommutative law 4A + (B + C) = ( A + B ) + CA. (B. C) = (A. B). Cassociative law 5A + (B. C) = ( A + B ). (A + C )A. (B + C) = ( A. B ) + (A. C )distributive law 6A + A = AA. A = Aindempotency 7A + 1 = 1A. 0 =0null Elements 8A + (A.B) = AA. (A+B) = Acovering 9A + (A’.B) = A + BA. (A’+B ) = A. Bminimization 10(A + B)’ = A’. B’(A. B)’ = A’ + B’de Morgan’s theory 11A. B + A. B’ = A(A + B). (A + B’) = Acombining 12A.B + A’.C + B.C = A.B + A’.C(A+B).(A’+C). (B+C)= (A+B).(A’+C)consensus

He/She can simplify boolean equations 5  The complex logic equations can be reduced or simplified to obtain a simpler form  The more simple logic equations the more economic and small electronic circuits QUESTION-1. Simplify the logic equation below (A+B). (A+C) = ? = A.A + A.C + B.A + B.C = A + A.C + A.B + B.C = A(1+C+B) + B.C = A + B.C QUESTION-2. Simplify the logic equation below A+ B.A= ? = A (1+B) = A

He/She can simplify boolean equations 6 QUESTION-3. Simplify the logic equation below A. (B+A)= ? = A.B + A.A = A (B+1) =A (1) =A QUESTION-4. Simplify the logic equation below A+ (A’.B)= ? = ((A+ (A’.B)’)’ = A’. ((A’.B)’)’ = A’. (A’’+B’)’ = A’. (A+B’)’ = A’.A + (A’.B’)’ = 0 + (A+B) = A+B

He/She can simplify boolean equations 7 QUESTION-5. Simplify the logic equation below A. (A’+B)= ? = A.A’ + A.B = 0 +A.B =A.B QUESTION-6. Simplify the logic equation below A’.B + A + A. B= ? = A’. B + A (1+B) = A’. B + A = (A’.B + A)’’ = ((A’.B)’.A’)‘ = ((A+B’).A’)’ = (A.A’ + A’.B’ )’ = (0 + A’.B’)’ = (A’. B’)’ = A + B

He/She can simplify boolean equations 8 QUESTION-7. Simplify the logic equation below A’.B’.C + A’.B.C + A.B’= ? =A’.C (B+B’) + A.B’ = A’.C (1) + A.B’ =A’.C + A.B’ QUESTION-6. Simplify the logic equation below A’.B’.C + A’.B’.C + A.B.C’= ? = A’.B’ (C +C’) + A.C’(B+B’) = A’.B’ (1) + A.C’(1) = A’.B’ + A.C’

He/She can construct truth tables 9 A truth table is a breakdown of a logic function by listing all possible values the function can attain. Such a table typically contains several rows and columns, with the top row representing the logical variables and combinations, in increasing complexity leading up to the final function. Let the number of input parameters as n. So there would be 2 n combinations. For example, for 2 inputs the table shows 2 2= 4 states.. A and B input variables A+B and A. B

He/She can construct truth tables 10 All combinations for input A and Input B

He/She can construct truth tables 11 Truth table proves that (A+B)’ = A’.B’Truth table proves that (A.B)’ = A’+B’

He/She can construct truth tables 12 Truth table proves that A+A.B = A

He/She can construct truth tables 13 Truth table proves that A.(B+C) = A.B + A.C

He/She can do arithmetic operations on various radix 14