Practice Questions for CSIR NET Group Theory : Sets and Relations II

21. If two sets \(A\) and \(B\) are having 99 elements in common, then the number of elements common to each of the sets \(A \times B\) and \(B \times A\) is

(a.) \(2^n\)
(b.) \(99^2\)
(c.) 100
(d.) 18

Explanation for Question 21:
\[ \begin{align*} A \times B \cap B \times A &= \{(x, y) \mid x, y \in A \cap B\} \\ \Rightarrow |A \times B \cap B \times A| &= 99 \times 99 = 99^2 \end{align*} \]

22. The relation "less than" in the set of natural numbers only is:

(a.) Only symmetric
(b.) Only transitive
(c.) Only reflexive
(d.) Equivalence relation

Explanation for Question 22:
a) \(a R b\) if and only if \(a + b\) is even, which means both \(a\) and \(b\) are either even or odd. Therefore, \(R\) is reflexive, symmetric, and transitive.
b) \(a R b\) if and only if \(a - b\) is even, which again means both \(a\) and \(b\) are either even or odd. Therefore, \(R\) is also reflexive, symmetric, and transitive.
c) As mentioned in the solution, \(1 < 2\), but \(2 \nless 1\), so \(R\) is not symmetric.
d) \(a R b\) if and only if \(a = b\), which implies that \(R\) is reflexive, symmetric, and transitive.

23. For real numbers \(x\) and \(y\), we write \(x R y \Leftrightarrow x-y+\sqrt{2}\) is an irrational number. Then the relation \(R\) is

(a.) Reflexive
(b.) Symmetric
(c.) Transitive
(d.) None of these

Explanation for Question 23:
\(x - x + \sqrt{2} = \sqrt{2} \in \mathbb{Q}^c\), so \(R\) is reflexive.
However, \(3 - \sqrt{2} + \sqrt{2} = 2\sqrt{2} - 3 \in \mathbb{Q}^c\), but \(3 - \sqrt{2} + \sqrt{2} = 3 \notin \mathbb{Q}^c\). So, \(R\) is not symmetric.
While \((0 - 2\sqrt{2} + \sqrt{2}, 2\sqrt{2} - \sqrt{2} + \sqrt{2})\) is in \(\mathbb{Q}^c\), \((0 - \sqrt{2} + \sqrt{2})\) is not in \(\mathbb{Q}^c\). Thus, \(R\) is not transitive.

24. The relation "is a subset of" on the power set \(P(A)\) of a set \(A\) is

(a.) Symmetric
(b.) Anti-symmetric
(c.) Equivalence relation
(d.) None of these

Explanation for Question 24:
If \(a \in A\) and \(A - \{a\} = B\), it implies \(B \subseteq A\), but \(A \nsubseteq B\). So, \(R\) is not symmetric, and therefore, it's not an equivalence relation.
Antisymmetric- \(A \nsubseteq B\) and \(B \subseteq A\) then we get \(B = A\).

25. The relation \(R\) defined on the set \(A = \{1,2,3,4,5\}\) by \(R = \{(x, y): |x^2-y^2|<16\}\) is given by:

(a.) \(\{(1,1),(2,1),(3,1),(4,1),(2,3)\}\)
(b.) \(\{(2,2),(3,2),(4,2),(2,4)\}\)
(c.) \(\{(3,3),(4,3),(5,4),(3,4)\}\)
(d.) None of these

Explanation for Question 25:
\((4^2 - 4^2) = 0 < 16\), which means \((4,4) \in R\).

26. Which of the following is not an equivalence relation in \(\mathbf{Z}\)?

(a.) \(a R b \Leftrightarrow a+b\) is an even integer
(b.) \(a R b \Leftrightarrow a-b\) is an even integer
(c.) \(a R b \Leftrightarrow a
(d.) \(a R b \Leftrightarrow a=b

Explanation for Question 26:
a) \(a R b\) if and only if \(a + b\) is even, which implies that both \(a\) and \(b\) are either even or odd. So, \(R\) is reflexive, symmetric, and transitive.
b) \(a R b\) if and only if \(a - b\) is even, which again implies that both \(a\) and \(b\) are either even or odd. Therefore, \(R\) is also reflexive, symmetric, and transitive.
c) The solution mentions that \(2 < 3\), but \(3 \nless 2\), which implies that \(R\) is not symmetric.
d) \(a R b\) if and only if \(a = b\), so \(R\) is reflexive, symmetric, and transitive.

27. Let \(X = \{1,2,3,4,5\}\) and \(Y = \{1,3,5,7,9\}\). Which of the following are not relations from \(X\) to \(Y\)?

(a.) \(R_1 = \{(x, y): y=x+2, x \in X\}\)
(b.) \(R_2 = \{(1,1),(2,1),(3,3),(4,3),(5,5)\}\)
(c.) \(R_3 = \{(1,1),(1,3),(3,5),(3,7),(5,7)\}\)
(d.) \(R_4 = \{(1,3),(2,5),(2,4),(7,9)\}\)

Explanation for Question 27:
Note that the relation from set \(A\) to set \(B\) is a subset of \(A \times B\). The solution mentions that options b, c, and d are not subsets of \(A \times B\).

28. Let \(R\) be a relation over the set \(N \times N\) and it is defined by \((a, b) R (c, d) \Rightarrow a+d=b+c\). Then \(R\) is:

(a.) Reflexive only
(b.) Symmetric only
(c.) Transitive only
(d.) An equivalence relation

Explanation for Question 28:
\(a + b = b + a \Rightarrow -(a, b) R (a, b)\), so \(R\) is reflexive.
To check symmetry If \(a+d=b+c\) (ie. \((a, b) R(c, d)\) then \(c+b= d+a\) \((c, d) R(a, b).\) So that R is transitive as well.

29. Let \(n\) be a fixed positive integer. Define a relation \(R\) on the set \(Z\) of integers by, \(a R b \Leftrightarrow n \mid(a-b)\). Then \(R\) is

(a.) Reflexive
(b.) Symmetric
(c.) Transitive
(d.) Equivalence

Explanation for Question 29:
\(a R b\) if and only if \(n \,|\, a - b\), which means \(a\) is congruent to \(b\) modulo \(n\). Therefore, \(R\) is an equivalence relation.

30. Which of the following is not an equivalence relation in \(\mathbf{R}\)?

(a.) \(x \leq y\) for all \(x, y \in \mathbb{R}\)
(b.) \(x-y\) is an irrational number
(c.) \(x-y\) is divisible by 3
(d.) \(x-y\) is a perfect square

Explanation for Question 30:
a) \(x \leq x\) for all \(x \in \mathbb{R}\), so \(R\) is reflexive. However, \(3 \leq 2\), but \(2 \nleq 3\), so \(R\) is not symmetric.
b) \(x - x = 0 \notin \mathbb{Q}^c\), so \(R\) is not reflexive.
c) \(3 \,|\, x - y\) if and only if \(x\) is congruent to \(y\) modulo 3, which makes \(R\) an equivalence relation.
d) The solution mentions \(7 - 6 = 1^2\), but \(7 - 5 = 2\), which implies \(R\) is not transitive. The explanation is incomplete for the last part of the question.

31. Define an equivalence relation \(\sim\) on \(\mathbb{R}\) as follows: given \(x, y \in \mathbb{R}, x \sim y\) if and only if \(x-y\) is a rational number. Then

(a.) Given \(x\), there are only finitely many \(y\) such that \(y \sim x\)
(b.) The Quotient set of \(\sim\) is infinite
(c.) Given \(x\) irrational number, there are only finitely many \(y\) such that \(y \sim x\)
(d.) None of these

Explanation for Question 31:
\(x \sim y\) if and only if \(x - y \in \mathbb{Q}\), which means there are infinitely many \(y\) such that \(y \sim x\).

32. Consider the following relations in \(\mathbf{Z}\), then which of the following is an equivalence relation?

(a.) \(m \sim n\) if and only if \(m n \geq 0\)
(b.) \(m \sim n\) if and only if \(m n > 0\)
(c.) \(m \sim n\) if and only if \(|m| = |n|\) where \(|n|\) denotes modulus of \(n\)
(d.) \(m \sim n\) if and only if \(m-n \leq 0\)

Explanation for Question 32:
Not applicable (NA) means that there's no information or explanation provided for this question.

33. The remainder obtained when \(16^{2016}\) is divided by 9 equals

(a.) 1
(b.) 2
(c.) 3
(d.) 7

Explanation for Question 33:
Not applicable (NA) means that there's no information or explanation provided for this question.

34. Let \(S\) be the set of all integers from 100 to 999 which are neither divisible by 3 nor divisible by 5. The number of elements in \(S\) is

(a.) 480
(b.) 420
(c.) 360
(d.) 240

Explanation for Question 34:
Not applicable (NA) means that there's no information or explanation provided for this question.