Please answer both parts of the question Decide whether the given structure form

Question

Please answer both parts of the question

Decide whether the given structure forms a ring. If it is not a ring, determine which of the ring axioms hold and which fail. You should work this out carefully for your own benet, but in your submitted solution, it is sufficient to simply list which axioms hold and which fail. (a) U is an arbitrary set, and R is the set of subsets of U. Addition and multiplication of elements of R are defined by the rules A+B- (AUB)(An B) and A B- AnB (Remark: you may find a Venn diagram helpful.) (b) R is the set of continuous function R R. Addition and multiplication are defined by the rules

Explanation / Answer

a. It's not a ring because it doesn't satisfy the addition and the multiplication property. E.g. In the addition property,A+B implies the collection of all the elements in the two sets ,but the L.H.S we have the union divided by intersection where it should just have been the union on the numerator and no denominator. Thus addition property fails.

b. The first property of addition is satisfied

(f+g)(x) =f (x)+g (x) ; reason being the whatever number of values x will take in the domain of f and g individually will remain the same as when it takes from the domain of f+g.

But the multiplicative property fails because

( f.g)(x)= f (g (x))

Might not hold true because for example if f (x)= x and g (x) =x

Then (f.g)(x)= f (x).g (x)= x22 butbut according to the problem

(f.g)(x)= f (g (x))= f (x)=x thus not equal to what it should be hence this structure is not a ring.

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