Set Theory
We will cover following topics
Completeness Of Real Line
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A metric space (X,d) in which every Cauchy sequence converges to an element of X is called complete, where d is the distance metric.
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The real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers.
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The rational numbers, on the other hand, are not complete under the usual distance metric. For example, the sequence defined by x0=1, xn+1=(xn+2xn)/2 consists of rational numbers but it converges to √2, which is irrational.
PYQs
Real number system as an ordered field
1) Suppose R be the set of all real numbers and f:R→R is a function such that the following equations hold for all x, y∈R:
f(x+y)=f(x)+f(y)
f(xy)=f(x)f(y)
Show that that ∀c∈R, either f(x)=0 or f(x)=x.
[2018, 20M]
2) Show that every open subset of R is a countable union of disjoint open intervals.
[2013, 13M]
3) Show that a bounded infinite subset of R must have a limit point.
[2009, 15M]
4) Let
T={1n,n∈N}∪{1+32n,n∈N}∪{6−13n,n∈N}.
Find derived set T′ of T. Also find Supremum of T and greatest number of T.
[2008, 6M]
5) Given a positive real number a and any natural number n, prove that there exists one and only one positive real number ξ such that ξn=a.
[2007, 20M]
6) If a continuous function of x satisfies the functional equation f(x+y)=f(x)+f(y), then show that f(x)=αx where α is a constant.
[2003, 12M]