Practice Questions for CSIR NET Group Theory : Basics of Group Theory II

16. Choose the false statement:

(a.) If * is a commutative binary operation on set \(S\) then \(a^*(b^*c)=(b^*c)^*a\) for all \(a, b, c \in S\).
(b.) Every binary operation defined on a set having exactly one element is both commutative and associative.
(c.) A binary operation on set \(S\) assigns exactly one element of \(S\) to each ordered pair of elements of \(S\).
(d.) A binary operation on set \(S\) may assign more than one element of \(S\) to some ordered pair of elements of \(S\).

Explanation for Question 16:
a) \[ \begin{aligned} a *(b * c) & =a * d \quad \text { (for } b * c=d) \\ & =d * a \\ & =(b * c) * a \end{aligned} \] b) \(S=\{a\}\) then \(S\) is commutative and associative as \(a * a=a\). c) & d) A binary operation is well defined if for each \(a, b \in S\), \(a * b\) is well defined.

17. On \(\mathbb {Z}^*\), define * by \(a * b=c\), where \(c\) is at least 5 more than \(a+b\), then,

(a.) * is not a binary operation.
(b.) * is a non-commutative binary operation.
(c.) * is a commutative binary operation.
(d.) * is an associative binary operation.

Explanation for Question 17:
\[ \begin{aligned} & a * b=a+b+a+5 \ \& \ a * b=a+b+6 \\ & \text { but } a+b+5 \neq a+b+6 \\ & \Rightarrow * \text { is not a binary operation } \end{aligned} \]

18. \(S=\{a, b, c\}\) and \( * \) is defined as shown in the table below:

\( * \)	\( a \)	\( b \)	\( c \)
\( a \)	\( a \)	\( b \)	\( c \)
\( b \)	\( b \)	\( d \)	\( c \)
\( c \)	\( c \)	\( b \)	\( a \)

then * is

(a.) Commutative
(b.) Associative
(c.) Not a binary operation
(d.) Binary operation and \(a\) is the identity element

Explanation for Question 18:
Given: \(b * b=d \notin S\) \[ \begin{aligned} & \Rightarrow * \text { is not a binary operation } \end{aligned} \]

19. In a set \(\mathbb{R}\) of real numbers, * is defined as \(a^*b=a+2b\), then choose the incorrect

(a.) * is commutative
(b.) * is associative
(c.) * is not a binary operation
(d.) None of these

Explanation for Question 19:
\[ \begin{aligned} & a *(b * c)=a *(b+2 c)=a+2 b+4 c \\ & (a * b) * c=(a+2 b) * c=a+2 b+2 c \end{aligned} \] \(\Rightarrow *\) is not associative but is a binary operation

20. On the set of real numbers \(\mathbb{R}\), \(^*\) is defined as \(a * b=\frac{n a+b}{n+1}\) where \(n \neq 1,-1\), then \(*\) is

(a.) Not a binary operation
(b.) Commutative binary operation
(c.) Associative binary operation
(d.) None of these

Explanation for Question 20:
For a fixed \(n\) define \(a * b=\frac{n a+b}{n+1} \in \mathbb{R}\), * is a binary operation. \[ \begin{aligned} & a * b=\frac{n a+b}{n+1} \\ & b * a=\frac{n b+a}{n+1} \end{aligned} \] \(\Rightarrow *\) is not commutative \[ \begin{gathered} a *(b * c)=\frac{n a+(b * c)}{n+1}=\frac{n a+\left(\frac{n b+c}{n+1}\right)}{n+1} \\ =\frac{(n+1) n a+n b+c}{(n+1)^{2}} \\ (a * b) * c=\frac{n(n a+b)+(n+1) c}{(n+1)^{2}} \end{gathered} \] \(\Rightarrow *\) is not associative

21. In \(P(\mathbb{N})\), let us define * such that \(A^* B=A \cap B\), then *

(a.) Is not a binary operation.
(b.) Is a commutative binary operation.
(c.) Is an associative binary operation.
(d.) None of these

Explanation for Question 21:
\(A-B\) is a subset of \(\mathbb{N} \Rightarrow A * B\) is a binary operation Let \(A=\{1,2\}, B=\{2,3\}\) \[ A-B=\{1\} \ \& \ B-A=\{3\} \] \(\Rightarrow\) is not commutative \[ \begin{aligned} & A=\{1,2\}, B=\{1\}, C=\{2\} \in P(\mathbb{N}) \\ & A *(B * C)=A-(B-C)=A-\{1\}=\{2\} \\ & (A * B) * C=(A-B)-C=\{2\}-C=\phi \\ & \Rightarrow * \text { is not associative } \end{aligned} \]

22. Let \(H(\mathbb{C})\) be the set of all matrices over the field of complex numbers with component-wise multiplication. Then

(a.) Is a binary operation.
(b.) Is not a binary operation.
(c.) Is an associative binary operation.
(d.) None of these

Explanation for Question 22:
Same as Question 15.

23. In \(\mathbb{Q}\), the set of all rational numbers with multiplication then,

(a.) Is not a binary operation.
(b.) Is an associative binary operation.
(c.) Is an associative and commutative binary operation.
(d.) None of these

Explanation for Question 23:
\[ \begin{aligned} & \text { Let } a, b \in \mathbb{Q} \Rightarrow b^{-1}=\frac{1}{b} \\ & a * b^{-1}=\frac{a}{b} \in \mathbb{Q} \quad \& \ a \cdot b=b \cdot a \end{aligned} \] \(\Rightarrow \mathbb{Q}\) is an abelian group. \(\Rightarrow\) * is associative and commutative.

24. Let \(A=\{1,2,3,4,5\}\) then which of the following is not a binary operation

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

Explanation for Question 24:
Every binary operation is a function from \(A \times A\) to \(A\).

25. Choose the correct statement/statements

(a.) Every binary operation is a relation.
(b.) Every function is a binary operation as well as a relation.
(c.) Every relation is a binary operation.
(d.) All of the above.

Explanation for Question 25:
Every binary operation is a relation from \(A \times A\) to \(A\).
\(f: \mathbb{R} \rightarrow \mathbb{R}\) cannot be a binary operation. Similarly, a relation \(\mathbb{R} \rightarrow \mathbb{R}\) is not a binary operation.

26. Let \(|A|=5\), then the number of binary operations on \(A\) is

(a.) \(5^5\)
(b.) \(5^{(5^5)}\)
(c.) \(5^{25}\)
(d.) \(5^4\)

Explanation for Question 26:
Number of \(b \cdot o =|A|^{|A \times A|}=5^{5^{2}}=5^{25}\).

27. Let us define * on \(\mathbb{Z}\) such that \(a * b = a - b\), then choose the incorrect statement:

(a.) * is commutative.
(b.) * is associative.
(c.) * is not a binary operation.
(d.) None of these

Explanation for Question 27:
\(a-b \in \mathbb{Z}\) for \(a, b \in \mathbb{Z}\). \(\Rightarrow *\) is a binary operation.
\(a-b \neq b-a\) \(\Rightarrow\) not commutative \[ (1-2)-3=-4 \ \& \ 1-(2-3)=1-(-1)=2 \] \(\Rightarrow\) not associative

28. Which of the following is a binary operation on the corresponding set:

(a.) \(a * b = ab + 1\) on \(\mathbb{Q}\)
(b.) \(a^* b = 2^{ab}\) on \(\mathbb{Z}^+\)
(c.) \(a^* b = a^b\) on \(\mathbb{Z}^+\)
(d.) \(a^* b = \frac{a+b}{2}\) on \(\mathbb{Z}\)

Explanation for Question 28:
a) \(a, b \in Q, a b+1 \in \mathbb{Q}\)
b) \(a, b \in \mathbb{Z}^{+}, \quad 2^{a b} \in \mathbb{Z}^{+}\)
c) \(a, b \in \mathbb{Z}^{+}, a^{b} \in \mathbb{Z}^{+}\)
d) \(0,1 \in \mathbb{Z}\) but \(\frac{0+1}{2}=\frac{1}{2} \notin \mathbb{Z}\)

29. In the set \(P(\mathbb{N})\), define \(A^* B = A \cap B\), then *

(a.) Is not a binary operation.
(b.) Is a commutative binary operation.
(c.) Is an associative binary operation.
(d.) Is a non-commutative binary operation.

Explanation for Question 29:
\(A \cap B \in P(\mathbb{N}) \Rightarrow\) * is a binary operation.
\(A \cap B=B \cap A\) and \(A \cap(B \cap C)=A \cap B \cap C=(A \cap B) \cap C\).
\(\Rightarrow\) * is commutative and associative.

30. The orders of \(a\) and \(x\) in a group are respectively 3 and 4. Then the order of \(x^{-1}ax\) is

(a.) 3
(b.) 4
(c.) 6
(d.) 12

Explanation for Question 30:
\[ \begin{aligned} o\left(x a x^{-1}\right) & =o(a) \quad(\text{Refer Question 13}) \\ & =3 \end{aligned} \]