Practice Questions for NET JRF Real Analysis Assignment: Countability of Sets - I

1. Let \(F\) be the set of all functions from \([0,1]\) to \([0,1]\) itself. If \(\operatorname{card}(F) = f\) then

(a.) \(F\) is similar to \([0,1]\)
(b.) \(f\) is less than the cardinality of \([0,1]\)
(c.) \(f > c\) where the cardinality of \([0,1]\) is \(c\)
(d.) \(F\) is countable

Explanation for Question 1:
Let \(S = \{f \ | \ f:A \rightarrow B\} =\) The set of all functions from \(A\) to \(B\). Then \(|S| = |B|^{|A|}\). In this case, we have \(c^c = 2^c > c\). This completes the explanation for Question 1.

2. If \(f: A \rightarrow B\) is a one-to-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

Explanation for Question 2:
If \(A\) is countable, it implies that \(f(A) \subset B\) is countable. This completes the explanation for Question 2.

3. Let \(A\) be an infinite set of disjoint open subintervals of \((0,1)\). Let \(B\) be the power set of \(A\). Then

(a.) The cardinality of \(A\) and \(B\) are equal
(b.) \(A\) is similar to \((0,1)\)
(c.) \(B\) is uncountable
(d.) The cardinality of \(B\) is greater than the cardinality of \(A\)

Explanation for Question 3:
From every element of \(A\), which is an open interval, we can select a unique rational representative since all subintervals are disjoint and non-trivial. Thus, \(|A| \leq |\mathbb{Q}|\), hence \(A\) is countable. Also, \(A\) is infinite implies that \(|A| = \aleph_0\). Then \(|P(A)| = 2^{\aleph_0} = c\). This completes the explanation for Question 3.

4. Match the following List-I with List-II and choose the correct:

List-I

A. Countable

B. Uncountable

C. Empty set

List-II

1. Set of transcendental elements in \(\mathbb{R}\)

2. Set of all functions from \(\mathbb{Z}_2 = \{0,1\}\) to \(\mathbb{N}\)

3. \(\{n \in \mathbb{N}: \sqrt{n+1}-\sqrt{n}\) is rational\(\}\)

(a.) 2 3 1
(b.) 1 2 3
(c.) 3 2 1
(d.) 2 1 3

Explanation for Question 4:
1. The set of all algebraic numbers is countable in \(\mathbb{R}\), thus the set of transcendental numbers is a complement of a countable set in \(\mathbb{R}\), hence uncountable. 2. Set of all functions has cardinality \(\aleph_0^2 = \aleph_0\) in this case, hence countable. 3. Any two consecutive natural numbers can't be perfect squares, thus the set is empty. This completes the explanation for Question 4.

5. Let \(A\) be the set of lines passing through the origin and having a slope that is an integral multiple of \(\frac{\pi}{12}\). Then

(a.) \(A\) is similar to \(\mathbb{R}\)
(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 \(\mathbb{R}\)

Explanation for Question 5:
For every integer, we have a line. Thus, there are countably infinite such lines. This completes the explanation for Question 5.

6. Consider the following statements:

1. The set of all finite subsets of the natural numbers is countable

2. The set of all polynomials with integer coefficients is countable

Choose the correct answer:

(a.) Only 1 is true
(b.) Only 2 is true
(c.) Both 1 and 2 are true
(d.) Both are false

Explanation for Question 6:
1. True statement. 2. For any infinite set \(A\), \(|A \times A| = |A|\). This completes the explanation for Question 6.

7. Let \(A\) and \(B\) be infinite sets. Let \(f\) be a map from \(A\) to \(B\) such that the collection of pre-images of any non-empty subset of \(B\) is non-empty. Which of the following statements is incorrect?

(a.) If \(A\) is countable, then \(B\) is countable
(b.) Such map \(f\) is always onto
(c.) \(A\) and \(B\) are similar
(d.) \(B\) may be countable even if \(A\) is not countable

Explanation for Question 7:
As the pre-image of every non-empty set is non-empty, we can use singleton sets in \(B\) to show that every element in \(B\) has an inverse image, thus \(f\) is an onto map. Implies that \(|A| \geq |B|\). Therefore, a, b, and d are true statements. This completes the explanation for Question 7.

8. If \(f\) is a function with domain \(A\) and range \(B\), then which of the following is correct?

(a.) If \(B\) is countable, then \(A\) is countable
(b.) If \(A\) is countable, then \(B\) is countable
(c.) If \(A\) is uncountable, then \(B\) is uncountable
(d.) All of the above

Explanation for Question 8:
Note that for any set \(A\), \(|f(A)| = |B| \leq |A|\). This completes the explanation for Question 8.

9. Which of the following statements is correct?

(a.) The set of rational numbers in any interval of finite length is countable
(b.) The set of irrational numbers in any interval of finite length is countable
(c.) Every subset of an uncountable set is uncountable
(d.) All of the above

Explanation for Question 9:
For a, since \(\mathbb{Q}\) is countable, thus every subset of \(\mathbb{Q}\) is countable. As a non-trivial interval is uncountable, it is a union of countable rationals and uncountable irrationals. Thus b is false. For c, we can consider an empty set for any set to be a countable set. This completes the explanation for Question 9.

10. Select the correct statement:

1. \(\phi \neq S\) is a countable set

2. There exists a surjection from \(\mathbb{N}\) onto \(S\)

3. There exists an injection from \(S\) into \(\mathbb{N}\)

(a.) Statement 1 implies statements 2 and 3, but not conversely
(b.) Statements 2 and 3 imply statement 1, but not conversely
(c.) Statement 1 implies either statement 2 or statement 3, but not both
(d.) All three statements are equivalent

Explanation for Question 10:
There exists a surjection from \(\mathbb{N}\) to \(S\) implies that \(|\mathbb{N}| \geq |S|\). If there exists an injection (1-1) map from \(S\) to \(\mathbb{N}\), it implies that \(|S| \leq |\mathbb{N}|\). Both imply that \(S\) is countable and thus are equivalent. This completes the explanation for Question 10.