sam2 - n = 2 8 Boolean Algebra State and prove DeMorgan’s...

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 satisfies
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