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**topic Elementary Set Theory and Countability**of the

**Chapter Real Analysis.**These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve

**IIT JAM Mathematics Previous Year Question Paper**. And Don’t Forget to Shore with Your Friends.

**IIT JAM Mathematics Test Series 2023 : Real Analysis – Elementary Set Theory and Countability**

**01. Let F be the set of all functions from [0,1] to [0.1] itself. If ****card(F) ****= ****f**** then**

(a) F is similar to [0,1]

(b) *f* is tess than the cardinality of [0,1]

(c) *f* > c where cardnality of [0,1] is c

(d) F is countable

**02.If ****f:A ****→ ****B**** is a one-one map and A is countable. Then which is correct**

(a) B is countable.

(b) B is uncountable

(c) There exists a subset of B which is countable

(d) None of these.

**03. Let A be an infinite set of disjoint open sub intervals of (0,1). Let B be the power set of A. Then-**

(a) Cardinality of A & B are equal

(b) A similar to (0, 1)

(c) B similar to (0, 1)

(d) A & B both are uncountable

**04. Let A be the set of ines passing through the origin and slope is integral multiple of Then**

(a) A is similar to R

(b) A is countably infinite

(c) A is similar to the set of months in a year

(d) A is similar to the power set of R.

**05. Consider the foliowing statements:****A.The set of all finite subsets of the natural numbers is countable****B. The set of all polynomials with integer coefticients is countable**

**Choose the correct answer:**

(a.) Only A is true

(b.) Only B is true

(c.) Bothl and B are true

(d.) Both are false

**06. Let A & B are infinite sets. Let ****f**** is a map from A to B such that the collection of pre images of any non-empty subset of B is non empty. Then choose the incorrect**

(a) If A is countable then B is countable

(b) Such map *f* is always onto

(c) A & B are similar

(d) B may be countable even if A is not countable

**07. If f be a function with domain A and range B then which of following is correct**.

(a) B countable =>A countable

(b) A countable => B countable

(c) A uncountable => B uncountable

(d) All of above

**08. Which of the following is correct?**

(a) The set of rational numbers in any interval of finite length is countable.

(b.) The set of irational nunbers in any interval of finite length is countable.

(c.) Every subset of uncountable set Is uncountable.

(d.) All of above.

**09. Read the following statements****A. Sis a countable set****B. There exists a surjection of N onto S.****C. There exists an injection of S into N**

Codes:

(a) A implies B & C but not conversely

(b) B & C imply A but not conversely

(c) A implies either of B & C but not both.

(d) All the three statements are equivalent.

**10. Which set is countable**

(a) The set of all polynomials with real coefficients

(b) The set of all subsets of a countable infinite set

(c) The set A-B where A is uncountable but B is countable

(d) The set of all finite subsets of N;T the set of natural number)

**11. If F be set of all function defined on In ={1,2,3,….., n}; n∈N with range B ⊆ I (set of positive integer) Then**

(a) F is countable.

(b) F is uncountable

(c) F is infinite

(d) F is countable if B is finite

**12. Select the correct statements****A. Every countable set is similar to N****B. The set of all disjoint intervałs is not similar to the set of real nụmbers****B. The power set of N is sinilar to the set of real numbers.**

**Codes:**

(a) A and B only

(b) B and C only

(c) A and C only

(d) All of these

**13. How many statements is/are false?****A. Cardinality of [0,1]×[0,1] is the same as cardinality of R****B. Cardinality of R is the same as the cardinality of irrationals****C. Cardinality of R is the same as the cardinality of C**

(a) Zero

(b) One

(c) Two

(d) Three

**14. Let A be an uncountable subset of R and B be a proper infinite subset of N. Define f:B → A such that f is one-one, then**

(a) A and A — B are similar

(b) A and A — f(B) are similar

(c) R and R — f(B) are similar

(d) All of the above are correct

**15. Let P _{n} be the set of all polynomials of degree n with integral coefficients. The P_{n} is**

(a) A finite set with n^{n} elements

(b) A finite set

(c) A countabie set

(d) An uncountable set

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*Answer Key* (if you find any answer wrong, feel free to comment below)

*Answer Key*

01. | (c) | 02. | (c) | 03. | (c) | 04. | (b) | 05. | (c) |

06. | (c) | 07. | (b) | 08. | (a) | 09. | (d) | 10. | (d) |

11. | (a) & (d) | 12. | (b) | 13. | (a) | 14. | (a), (b), (c), & (d) | 15. | (a) |

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