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solutionshw1-554-2010

# solutionshw1-554-2010 - Solutions homework 1 Extra Problem...

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Unformatted text preview: Solutions homework 1. Extra Problem 1 : First observe that F is closed under the addition and multiplication as defined. To check (A1) ( a + b ) + c = a + ( b + c ), note that this holds if two of the three elements equal 0, as then the sums on both sides equal 0. If all 3 elements are 1, then both sides equal 1, since 0 + 1 = 1 + 0. If e.g a = b = 1 and c = 0. Then the lhs equals 0 + 0 = 0 while the rhs equals 1 + 1 = 0. Similarly we can check the other combinations. Axiom (A2) holds clearly, and (A3) also obviously hold for 0, while from 1+1 = 0 it follows that- 1 = 1. Hence the additive axioms hold. Axiom (M1) holds clearly, as both sides are 0, except when a = b = c = 1 in which case both sides equal 1. Axioms (M2), (M3), (M4) are obvious. To see that (D) holds note that both sides are 0, when a = 0, and both sides are equal to b + c when a = 1. Hence F is a field. Extra Problem 2. a. True, Given x , take y = x + 1....
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