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pset1s

# pset1s - Massachusetts Institute of Technology Department...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.111 - Introductory Digital Systems Laboratory Problem Set 1 Solutions Issued : February 13, 2004 Problem 1 : Boolean Algebra Practice Problems (Problem 1 was not graded.) 1) a + 0 = a 2) a 0 = 0 3) a + a = 1 4) a + a = a 5) a + ab = a 1 ( + b ) = a 6) a + b a = ( a + a )( a + b ) = a + b 7) a a + b ) = a a + ab = ab ( 8) ab + b a = a b + a ) = b ( 9) ( a + b )( a + b ) = a a + b a + a b + b b = a + b a + b a = a 1 ( + b + b ) = a 10) a a + b + c + ...) = aa + ab + ac + ... = a + ab + ac + ... = a ( 11) f ( ab b a ) = a + b + ab = a + b , , 12) f ( a b a b ) = a + b + b a = a + b + a = 1 , , 13) f [ b a ( , ab ] ) = a + b + ( ab ) = a + b + a + b = 1 , 14) y + y y = y 15) xy + y x = y x + y ) = x ( 16) x + x y = x 1 ( + y ) = x 17) ( w + x + y + z ) y = y 18) ( x + y )( x + y ) = x 19) w + [ w + ( wx )] = w 20) x x + ( xy )] = x [ 21) ( x + x ) = x 22) ( x + x ) = 0 23) w + ( yz x w ) = w 1 ( + yz x ) = w 24) w ( wxyz ) = w w + x + y + z ) = w ( 25) xz + y x + zy = xz + y x 26) ( x + z )( x + y )( z + y ) = ( x + z )( x + y ) 27) x + y + z xy = x + y + z

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Problem 2: Karnaugh Maps and Minimal Expressions It is best to approach each expression by first simplifying it using Boolean algebra learned Problem 1. If you can put the expression into a product-of-sums or sum-of-
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pset1s - Massachusetts Institute of Technology Department...

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