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Page 1 Edurev123 Real Analysis 1. Real Number System 1.1 Show that every open subset of ?? is a countable union of disjoint open intervals. (2013 : 14 Marks) Solution: Let ?? =?? be an open and let ?? ??? . Then ?? is rational or irrational. If ?? is rational, define ?? ?? = ? ? ?? ?? ??? ??? ?? is open ?? (Union of all open intervals in ?? containing ?? ) Note that ?? ?? is simply the largest open interval containing ?? as it is union of disjoint open intervals (see figure below). If ?? is irrational, as ?? is open, ??>0 such that (?? -?? ,?? +?? )??? . As every interval contains a rational ??? ?(?? -?? ,?? +?? ) and then ?? ??? ?? . ???? ????? ??? ?? for some ?? ??? ??? , i.e., where ?? is a rational in ?? . ?????????????????????????????????????????????????????????????????? ?UI ?? ??????? But ?? ?? =?? ??? ?????????????????????????????????????????????????????????????????? = ?? ?? ?? ?? ??? ??? Page 2 Edurev123 Real Analysis 1. Real Number System 1.1 Show that every open subset of ?? is a countable union of disjoint open intervals. (2013 : 14 Marks) Solution: Let ?? =?? be an open and let ?? ??? . Then ?? is rational or irrational. If ?? is rational, define ?? ?? = ? ? ?? ?? ??? ??? ?? is open ?? (Union of all open intervals in ?? containing ?? ) Note that ?? ?? is simply the largest open interval containing ?? as it is union of disjoint open intervals (see figure below). If ?? is irrational, as ?? is open, ??>0 such that (?? -?? ,?? +?? )??? . As every interval contains a rational ??? ?(?? -?? ,?? +?? ) and then ?? ??? ?? . ???? ????? ??? ?? for some ?? ??? ??? , i.e., where ?? is a rational in ?? . ?????????????????????????????????????????????????????????????????? ?UI ?? ??????? But ?? ?? =?? ??? ?????????????????????????????????????????????????????????????????? = ?? ?? ?? ?? ??? ??? And as the number of rationals in ?? are countable ?? can be writien as union of countable intervals. Also each of ?? ?? is disjoint as it being maximal if two ?? ?? overlap they will be same.Read More
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FAQs on Real Number System - Mathematics Optional Notes for UPSC
| 1. What are real numbers in the number system? |  |
Ans. Real numbers are all the numbers on the number line, including both rational and irrational numbers. They can be represented as decimals or fractions.
| 2. How are real numbers classified in the number system? | |
Ans. Real numbers are classified into rational numbers and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot be expressed as fractions.
| 3. Can real numbers be negative? | |
Ans. Yes, real numbers can be negative. They can be represented on the number line to the left of zero.
| 4. Are integers considered real numbers in the number system? | |
Ans. Yes, integers are considered real numbers. They are whole numbers that can be positive, negative, or zero.
| 5. How are real numbers used in practical applications? | |
Ans. Real numbers are used in various real-life applications such as measurements, calculations, and financial transactions. They are essential in fields like science, engineering, and economics.
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Jan 05, 2025
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Document Description: Real Number System for UPSC 2025 is part of Mathematics Optional Notes for UPSC preparation.
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