sam2 - n = 2. 8. Boolean Algebra: State and prove DeMorgans...

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CSE20 Exercise 1, February 12, 2010, 1. Residual Number System: Represent 38 with a residual number system of moduli ( m 1 , m 2 , m 3 )= (3, 5, 7). 2. Residual Number System: Suppose ( x %5 , x %7 , x %11) = (1 , 2 , 3), where symbol % denotes modulus operation. Find the smallest positive integer x that satis±es this system. 3. Residual Number System: Show the operation of 38 + 44 in a residual number system with moduli ( m 1 , m 2 , m 3 ) = (3 , 5 , 7). 4. Residual Number System: Show the operation of 19 × 15 in a residual number system with moduli ( m 1 , m 2 , m 3 ) = (5 , 13 , 14). 5. Residual Number System: State and prove the Chinese remainder theorm. 6. Boolean Algebra: Prove that for any a and b in the set B of a Boolean algebra, ( a + b )( a + b 0 ) = a . 7. Boolean Algebra: Prove general associativity holds for + in any Boolean algebra. For all n 1, a 1 + ( a 2 + ( a 2 + ( . . . + a n ))) = ((( a 1 + a 2 ) + a 3 ) + . . . ) + a n You may assume that associativity holds for
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Unformatted text preview: n = 2. 8. Boolean Algebra: State and prove DeMorgans laws. 9. Boolean Algebra: Show the operation tables for a Boolean algebra of four ele-ments. 10. Boolean Algebra: Simplify formula ( pq + r )( p + r )( q + r ). 11. Boolean Algebra: Express Boolean function E ( x, y, z ) = ( x + y )( xy ) ( x + y + z ) in sum-of-products form. 12. Boolean Algebra: Express Boolean function E ( x, y, z ) = xy + ( x + z ) + x y z in product-of-sums form. 13. Boolean Algebra: Prove or disprove the Boolean equation, ( a b + c )( a + b )( b + ac ) = a bc . 14. Boolean Algebra: Reduce the following to an expression of a minimal number of literals (4). abc d + ab c + bc d + ab c + acd + a bcd . 1...
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