Question

250 லிட்டரின் 12% என்பது 150 லிட்டரின் _______ க்குச் சமமாகும். *
1 point
10%
40%
20%
15%​

Answer 1

Answer:

Suppose that X and R∖X are both open. Pick a point a∈X and a point b∈R∖X. There’s no harm in assuming that a<b. Let A={x∈[a,b]:[a,x]⊆X}. A is bounded, so it has a least upper bound u. Clearly u≤b.

Now let ϵ>0, and consider the interval J=(u−ϵ,u+ϵ). Since u is the least upper bound of A, A∩(u−ϵ,u]≠∅, and therefore certainly J∩A≠∅. This shows that u can’t be in R∖X, since no open nbhd of u is a subset of R∖X, Thus, u∈A, and therefore u<b (since b∉A). Let v=min{u+ϵ,b}. Clearly v∉A, so there is some x∈[a,v)∖X. since [a,u]⊆X, we must have x∈[u,v)⊆[u,u+ϵ). Thus, J∖X≠∅, and J⊈X. Thus, u can’t be in X, either. Since u has to be somewhere, this is a contradiction, showing that X and R∖X can’t both be open.

Answer 2

Answer:

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