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#### Question 1:

Let '*' be a binary operation on N defined by
a * b = 1.c.m. (a, b) for all a, bN
(i) Find 2 * 4, 3 * 5, 1 * 6.
(ii) Check the commutativity and associativity of '*' on N.

a * b = 1.c.m. (a, b)

(i) 2 * 4 = 1.c.m. (2, 4)
= 4
3 * 5 = 1.c.m. (3, 5)
= 15
1 * 6 = 1.c.m. (1, 6)
= 6

(ii) Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is associative on N.

#### Question 2:

Determine which of the following binary operations are associative and which are commutative:
(i) * on N defined by a * b = 1 for all a, bN
(ii) * on Q defined by

(i) Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is associative on N.

(ii) Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is not associative on N.

#### Question 3:

Let A be any set containing more than one element. Let '*' be a binary operation on A defined by

a * b = b for all a, bA

Is '*' commutative or associative on A?

Commutativity:

Thus, * is not commutative on A.

Associativity:

Thus, * is associative on A.

#### Question 4:

Check the commutativity and associativity of each of the following binary operations:
(i) '*'. on Z defined by a * b = a + b + ab for all ab ∈ Z
(ii) '*'. on N defined by a * b = 2ab for all ab ∈ N
(iii) '*'. on Q defined by a * b = a − b for all ab ∈ Q
(iv) '⊙' on Q defined by a ⊙ b = a2 + b2 for all ab ∈ Q
(v) 'o' on Q defined by  for all ab ∈ Q
(vi) '*' on Q defined by a * b = ab2 for all ab ∈ Q
(vii) '*' on Q defined by a * b = a + ab for all ab ∈ Q
(viii) '*' on R defined by a * b = a + b − 7 for all ab ∈ R
(ix) '*' on Q defined by a * b = (a − b)2 for all ab ∈ Q
(x) '*' on Q defined by a * b = ab + 1 for all ab ∈ Q
(xi) '*' on N, defined by a * b = ab for all ab ∈ N
(xii) '*' on Z defined by a * b = a − b for all ab ∈ Z
(xiii) '*' on Q defined by $a*b=\frac{ab}{4}$ for all ab ∈ Q
(xiv) '*' on Z defined by a * b = a + b − ab for all ab ∈ Z
(xv) '*' on N defined by a * b = gcd(ab) for all ab ∈ N

(i) Commutativity:

Thus, * is commutative on Z.

Associativity:

Thus, * is associative on Z.

(ii) Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is not associative on N.

(iii) Commutativity:

Thus, * is not commutative on Q.

Associativity:

Thus, * is not associative on Q.

(iv) Commutativity:

Thus, $\odot$ is commutative on Q.

Associativity:

Thus, $\odot$ is not associative on Q.

(v) Commutativity:

Thus, o is commutative on Q.

Associativity:

Thus, is  associative on Q.

(vi) Commutativity:

Thus, * is not commutative on Q.

Associativity:

Thus, * is not associative on Q.

(vii) Commutativity:

Thus, * is not commutative on Q.

Associativity:

Thus, * is not associative on Q.

(viii) Commutativity:

Thus, * is commutative on R.

Associativity:

Thus, * is associative on R.

(ix) Commutativity:

Thus, * is commutative on Q.

Associativity:

Thus, * is not associative on Q.

(x) Commutativity:

Thus, * is commutative on Q.

Associativity:

Thus, * is not associative on Q.

(xi) Commutativity:

Thus, * is not commutative on N.

Associativity:

Thus, * is not associative on N.

(xii) Commutativity:

Thus, * not is commutative on Z.

Associativity:

Thus, * is not associative on Z.

(xiii) Commutativity:

Thus, * is commutative on Q.

Associativity:

Thus, * is associative on Q.

(xiv) Commutativity:

Thus, * is commutative on Z.

Associativity:

Thus, * is associative on Z.

Disclaimer : The answer given in the textbook is incorrect. The same has been corrected here.

(xv) Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is associative on N.

#### Question 5:

If the binary operation o is defined by aob = a + bab on the set Q − {−1} of all rational numbers other than 1, shown that o is commutative on Q − [1].

Thus, o is commutative on Q - {1}.

#### Question 6:

Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative.

Thus, * is not commutative on Z.

#### Question 7:

On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a , bZ. Prove that * is not associative on Z.

Thus, * is not associative on Z.

#### Question 8:

Let S be the set of all real numbers except −1 and let '*' be an operation defined by

a * b = a + b + ab for all a, bS.

Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.

Checking for binary operation:

Thus, * is a binary operation on S.

Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is associative on S.

Now,

#### Question 9:

On Q, the set of all rational numbers, * is defined by $a*b=\frac{a-b}{2}$, shown that * is no associative.

Thus, * is not associative on Q.

#### Question 10:

On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.

Commutativity:

Thus, * is not commutative on Z.

Associativity:

Thus, * is not associative on Z.

#### Question 11:

On the set Q of all ration numbers if a binary operation * is defined by $a*b=\frac{ab}{5}$, prove that * is associative on Q.

Thus, * is associative on Q.

#### Question 12:

The binary operation * is defined by $a*b=\frac{ab}{7}$ on the set Q of all rational numbers. Show that * is associative.

Thus, * is associative on Q.

#### Question 13:

On Q, the set of all rational numbers a binary operation * is defined by $a*b=\frac{a+b}{2}$.
Show that * is not associative on Q.

Thus, * is not associative on Q.

#### Question 14:

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b $-$ ab, for all a, b $\in$ S.
Prove that:
(i) * is a binary operation on S
(ii) * is commutative as well as associative.                                                                                                                                    [CBSE 2014]

We have,

S = R $-$ {1} and * is defined on S as a * b = a + b $-$ ab, for all ab $\in$ S

(i) It is seen that for each a, b $\in$ S, there is a unique element a + b $-$ ab in S

This means that * carries each pair (a, b) to a unique element a * b = a + b $-$ ab in S

So, * is a binary operation on S

(ii) Commutativity:

Thus, * is commutative on S.

Associativity:

Thus , * is associative on S.

So, * is commutative as well as associative.

#### Question 1:

Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, bI+.

Let e be the identity element in I+ with respect to * such that

Thus, 0 is the identity element in I+ with respect to *.

#### Question 2:

Find the identity element in the set of all rational numbers except −1 with respect to *defined by a * b = a + b + ab.

Let e be the identity element in Q$-$ {$-$1} with respect to * such that

Thus, 0 is the identity element in Q - {-1} with respect to *.

#### Question 3:

If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.

Let e be the identity element in Z with respect to * such that

Thus, 5 is the identity element in Z with respect to *.

#### Question 4:

On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.

Let e be the identity element in Z with respect to * such that

Thus, -2 is the identity element in Z with respect to *.

#### Question 1:

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b Z
(i) Show that '*' is both commutative and associative.
(ii) Find the identity element in Z.
(iii) Find the invertible elements in Z.

(i) Commutativity:

Thus, * is commutative on Z.

Associativity:

Thus, * is associative on Z.

(ii) Let e be the identity element in Z with respect to * such that

Thus, 4 is the identity element in Z with respect to *.

#### Question 2:

Let * be a binary operation on Q0 (set of non-zero rational numbers) defined by
.
Show that * is commutative as well as associative. Also, find its identity element if it exists.

Commutativity:

Thus, * is commutative on Qo.

Associativity:

Thus, * is associative on Qo.

Finding identity element:

Let e be the identity element in Z with respect to * such that

Thus, 5 is the identity element in Qo with respect to *.

#### Question 3:

Let * be a binary operation on Q − {−1} defined by
a * b = a + b + ab for all a, bQ − {−1}
Then,
(i) Show that '*' is both commutative and associative on Q − {−1}.
(ii) Find the identity element in Q − {−1}
(iii) Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element.

(i) Commutativity:

Thus, * is commutative on Q $-${$-$1}.

Associativity:

Thus, * is associative on Q $-$ {$-$1}.

(ii) Let e be the identity element in Q$-$ {$-$1} with respect to * such that

Thus, 0 is the identity element in Q $-$ {$-$1} with respect to *.

#### Question 4:

Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows :
(a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R.
(i) Show that '⊙' is commutative and associative on A
(ii) Find the identity element in A
(iii) Find the invertible elements in A.

(i) Commutativity:

Thus, $\odot$ is commutative on A.

Associativity:

#### Question 5:

Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by

.

(i) Show that 'o' is both commutative and associate.
(ii) Find the identity element in Q0.
(iii) Find the invertible elements of Q0.

(i) Commutativity:

Thus, o is commutative on Qo.

Associativity:

Thus, o is associative on Qo.

(ii) Let e be the identity element in Qo with respect to * such that

Thus, 2 is the identity element in Qo with respect to o.

#### Question 6:

On R − {1}, a binary operation * is defined by a * b = a + bab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.

Commutativity:

Thus, * is commutative on R $-$ {1}.

Associativity:

Thus, * is associative on R $-$ {1}.

Finding identity element:
Let e be the identity element in R $-$ {1} with respect to * such that

Thus, 0 is the identity element in R $-${1} with respect to *.

Finding inverse:

#### Question 7:

Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
(i) Show that '*' is both commutative and associative on A
(ii) Find the identity element in A
(iii) Find the invertible element in A.

#### Question 8:

Let * be the binary operation on N defined by
a * b = HCF of a and b.
Does there exist identity for this binary operation one N?

Let e be the identity element. Then,

We cannot find e that satisfies this condition.
So, the identity element with respect to * does not exist in N.

#### Question 9:

Let A$=$R$×$R and $*$ be a binary operation on defined by $\left(a,b\right)*\left(c,d\right)=\left(a+c,b+d\right).$ Show that $*$ is commutative and associative. Find the binary element for $*$ on A, if any.

We have,

A$=$R$×$R and $*$ is a binary operation on A defined by .

Now,

So, $*$ is commutative.

Also,

So, $*$ is associative.

Let (x, y) be the binary element for $*$ on A.

Hence, (0, 0) is the binary element for $*$ on A.

#### Question 1:

Construct the composition table for ×4 on set S = {0, 1, 2, 3}.

Here,

1 ${×}_{4}$ 1 = Remainder obtained by dividing 1 $×$1 by 4
= 1

0 ${×}_{4}$ 1 = Remainder obtained by dividing 0 $×$ 1 by 4
= 0

2 ${×}_{4}$ 3 = Remainder obtained by dividing 2 $×$ 3 by 4
= 2

3 ${×}_{4}$ 3 = Remainder obtained by dividing 3 $×$ 3 by 4
= 1

So, the composition table is as follows:

 ${×}_{4}$ 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1

#### Question 2:

Construct the composition table for +5 on set S = {0, 1, 2, 3, 4}.

Here,

1 ${+}_{5}$ 1 = Remainder obtained by dividing 1 + 1 by 5
= 2

3 ${+}_{5}$ 4 = Remainder obtained by dividing 3 + 4 by 5
= 2

4 ${+}_{5}$ 4 = Remainder obtained by dividing 4 + 4 by 5
= 3

So, the composition table is as follows:

 ${+}_{5}$ 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 5 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3

#### Question 3:

Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.

Here,

1 ${×}_{6}$ 1 = Remainder obtained by dividing 1 $×$ 1 by 6
= 1

3 ${×}_{6}$ 4 = Remainder obtained by dividing 3 $×$ 4 by 6
= 0

4 ${×}_{6}$ 5 = Remainder obtained by dividing 4$×$ 5 by 6
= 2

So, the composition table is as follows:

 ${×}_{6}$ 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 1 2 3 4 5 2 0 2 4 0 2 4 3 0 3 0 3 0 3 4 0 4 2 0 4 2 5 0 5 4 3 2 1

#### Question 4:

Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.

Here,

1 ${×}_{5}$ 1 = Remainder obtained by dividing 1 $×$ 1 by 5
= 1

3 ${×}_{5}$ 4 = Remainder obtained by dividing 3 $×$ 4 by 5
= 2

4 ${×}_{5}$ 4 = Remainder obtained by dividing 4 $×$ 4 by 5
= 1

So, the composition table is as follows:

 ${×}_{5}$ 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1

#### Question 5:

For the binary operation ×10 on set S = {1, 3, 7, 9}, find the inverse of 3.

Here,

1 ${×}_{10}$ 1 = Remainder obtained by dividing 1 $×$ 1 by 10
=1
3 ${×}_{10}$ 7 = Remainder obtained by dividing 3 $×$ 7 by 10
=1
7 ${×}_{10}$ 9 = Remainder obtained by dividing 7 $×$ 9 by 10
= 3

So, the composition table is as follows:

 ${×}_{10}$ 1 3 7 9 1 1 3 7 9 3 3 9 1 7 7 7 1 9 3 9 9 7 3 1

We observe that the elements of the first row are same as the top-most row.
So, $1\in S$ is the identity element with respect to ${×}_{10}$.

Finding inverse of 3:

From the above table we observe,
3 ${×}_{10}$ 7 = 1

So, the inverse of 3 is 7.

#### Question 6:

For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.

Finding identity element:
Here,

1 ${×}_{7}$ 1 = Remainder obtained by dividing 1 $×$ 1 by 7
= 1

3 ${×}_{7}$ 4 = Remainder obtained by dividing 3 $×$ 4 by 7
= 5

4 ${×}_{7}$ 5 = Remainder obtained by dividing 4$×$ 5 by 7
= 6

So, the composition table is as follows:

 ${×}_{7}$ 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 1 3 5 3 3 6 2 5 1 4 4 4 1 5 2 6 3 5 5 3 1 6 4 2 6 6 5 4 3 2 1

We observe that all the elements of the first row of the composition table are same as the top-most row.
So, the identity element is 1.

Also,
So, 3$-1$ = 5

#### Question 7:

Find the inverse of 5 under multiplication modulo 11 on Z11.

Here,
1 ${×}_{11}$ 1 = Remainder obtained by dividing 1 $×$ 1 by 11
= 1

3 ${×}_{11}$ 4 = Remainder obtained by dividing 3 $×$ 4 by 11
= 1

4 ${×}_{11}$ 5 = Remainder obtained by dividing 4 $×$ 5 by 11
= 9
So, the composition table is as follows:

 ${×}_{11}$ 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 1 3 5 7 9 3 3 6 9 1 4 7 10 2 5 8 4 4 8 1 5 9 2 6 10 3 7 5 5 10 4 9 3 8 2 7 1 6 6 6 1 7 2 8 3 9 4 10 5 7 7 3 10 6 2 9 5 1 8 4 8 8 5 2 10 7 4 1 9 6 3 9 9 7 5 3 1 10 8 6 4 2 10 10 9 8 7 6 5 4 3 2 1

We observe that the first row of the composition table is same as the top-most row.
So, the identity element is 1.

Also,

#### Question 8:

Write the multiplication table for the set of integers modulo 5.

Here,

1 ${×}_{5}$×5 1 = Remainder obtained by dividing 1$×$ 1 by 5
= 1

3 ${×}_{5}$ 4 = Remainder obtained by dividing 3 $×$4 by 5
= 2

${×}_{5}$4 = Remainder obtained by dividing 4 $×$ 4 by 5
= 1

So, the composition table is as follows:

 ${×}_{5}$ 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1

#### Question 9:

Consider the binary operation * and o defined by the following tables on set S = {a, b, c, d}.
(i)

 * a b c d a a b c d b b a d c c c d a b d d c b a

(ii)
 o a b c d a a a a a b a b c d c a c d b d a d b c

Show that both the binary operations are commutative and associatve. Write down the identities and list the inverse of elements.

(i) Commutativity:
The table is symmetrical about the leading element. It means * is commutative on S.

Associativity:

So, * is associative on S.

Finding identity element:
We observe that the first row of the composition table coincides with the top-most row and the first column coincides with the left-most column.
These two intersect at a.

So, a is the identity element.

Finding inverse elements:

(ii)  Commutativity:
The table is symmetrical about the leading element. It means that o is commutative on S.
Associativity:

So, o is associative on S.

Finding identity element:
We observe that the second row of the composition table coincides with the top-most row and the first column coincides with the left-most column.
These two intersect at b.

So, b is the identity element.

Finding inverse elements:

#### Question 10:

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as

Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.

Here,
1 * 1 =1+1                ($\because$ 1+1 $<$6 )
= 2
3 * 4 = 3 + 4 $-$6       ($\because$ 3 + 4  $>$6 )
= 7 $-$ 6
= 1
4 * 5 = 4 + 5$-$6          ($\because$ 4 + 5$>$6 )
= 9$-$ 6
= 3 etc.
So, the composition table is as follows:

 * 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4

We observe that the first row of the composition table coincides with the top-most row and the first column coincides with the left-most column.
These two intersect at 0.
So, 0 is the identity element .

Finding inverse:

#### Question 1:

Write the identity element for the binary operation * on the set R0 of all non-zero real numbers by the rule for all a, bR0.

Let e be the identity element in R0 with respect to * such that

Thus, 2 is the identity element in R0 with respect to *.

#### Question 2:

On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, bZ. Write the inverse of 4.

To find the identity element, let e be the identity element in Z with respect to * such that

Thus,$-$2 is the identity element in Z with respect to *.

Now,

#### Question 3:

Define a binary operation on a set.

Let A be a non-empty set. An operation * is called a binary operation on A, if and only if

#### Question 4:

Define a commutative binary operation on a set.

An operation * on a set A is called a commutative binary operation if and only if it is a binary operation as well as commutative, i.e. it must satisfy the following two conditions.

#### Question 5:

Define an associative binary operation on a set.

An operation * on a set A is called an associative binary operation if and only if it is a binary operation as well as associative, i.e. it must satisfy the following two conditions:

#### Question 6:

Write the total number of binary operations on a set consisting of two elements.

Number of binary operations on a set with n elements = ${n}^{{n}^{2}}$

#### Question 7:

Write the identity element for the binary operation * defined on the set R of all real numbers by the rule

Let e be the identity element in R with respect to * such that

Thus, $\frac{7}{3}$ is the identity element in R with respect to *.

#### Question 8:

Let * be a binary operation, on the set of all non-zero real numbers, given by

Write the value of x given by 2 * (x * 5) = 10.

#### Question 9:

Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.

As, e = 1 : 5 × 9 ≡ 1 (mod 11)

So, the inverse of 5 i.e. 5$-$1 = 9

#### Question 10:

Define identity element for a binary operation defined on a set.

Let * be a binary operation on a set A.

An element e is called an identity element in A with respect to * if and only if

#### Question 11:

Write the composition table for the binary operation multiplication modulo 10 (×10) on the set S = {2, 4, 6, 8}.

Here,

2 ${×}_{10}$ 4 = Remainder obtained by dividing 2 $×$ 4 by 10
= 8

4 ${×}_{10}$ 6 = Remainder obtained by dividing 4 $×$ 6 by 10
= 4

2 ${×}_{10}$ 8 = Remainder obtained by dividing 2 $×$ 8 by 10
= 6

3 ${×}_{10}$ 4 = Remainder obtained by dividing 3 $×$ 4 by 10
= 2
So, the composition table is as follows:

 ${×}_{10}$ 2 4 6 8 2 4 8 2 6 4 8 6 4 2 6 2 4 6 8 8 6 2 8 4

#### Question 12:

For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.

Here,

1 ${×}_{10}$ 1 = Remainder obtained by dividing 1 $×$ 1 by 10
= 1

3 ${×}_{10}$ 1 = Remainder obtained by dividing 3 $×$ 1 by 10
= 3

7 ${×}_{10}$ 3 = Remainder obtained by dividing 7 $×$ 3 by 10
= 1

3 ${×}_{10}$ 3 = Remainder obtained by dividing 3$×$ 3 by 10
= 9

So, the composition table is as follows:

 ${×}_{10}$ 1 3 7 9 1 1 3 7 9 3 3 9 1 7 7 7 1 9 3 9 9 7 3 1

We observe that the first row of the composition table coincides with the top-most row and the first column coincides with the left-most column.

These two intersect at 1.

So, the identity element is 1.

Also,
3 ${×}_{10}$ 7 = 1
3-1 = 7

#### Question 13:

For the binary operation multiplication modulo 5 (×5) defined on the set S = {1, 2, 3, 4}. Write the value of ${\left(3{×}_{5}{4}^{-1}\right)}^{-1}$.

Here,

1 ${×}_{5}$1 = Remainder obtained by dividing 1 $×$ 1 by 5
= 1

3 ${×}_{5}$ 4 = Remainder obtained by dividing 3 $×$ 4 by 5
= 2

${×}_{5}$ 4 = Remainder obtained by dividing 4 $×$ 4 by 5
= 1
So, the composition table is as follows:

 ${×}_{5}$   × 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1

We observe that the first row of the composition table coincides with the top-most row and the first column coincides with the left-most column.

These two intersect at 1.

Thus, 1 is the identity element.

#### Question 14:

Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.

Here,

1${×}_{5}$1 = Remainder obtained by dividing 1 $×$ 1 by 5
= 1

3${×}_{5}$4 = Remainder obtained by dividing 3 $×$ 4 by 5
= 2

4${×}_{5}$4 = Remainder obtained by dividing 4 $×$ 4 by 5
= 1

So, the composition table is as follows:

 ${×}_{5}$   × 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1

#### Question 15:

A binary operation * is defined on the set R of all real numbers by the rule

Write the identity element for * on R.

Let e be the identity element in R with respect to * such that

Thus, 0 is the identity element in R with respect to *.

#### Question 16:

Let +6 (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of $2{+}_{6}{4}^{-1}{+}_{6}{3}^{-1}.$

Here,

${+}_{6}$ 1 = Remainder obtained by dividing 1 + 1 by 6
= 2

3 ${+}_{6}$ 4 = Remainder obtained by dividing 3 + 4 by 6
= 1

4 ${+}_{6}$ 5 = Remainder obtained by dividing 4 + 5 by 6
= 3

So, the composition table is as follows:

 ${+}_{6}$ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4

We observe that the first row of the composition table coincides with the the top-most row and the first column coincides with the left-most column.
These two intersect at 0.

So, 0 is the identity element.

From the table,

#### Question 17:

Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.

Given: a * b = 3a + 4b − 2

Here,

4 * 5 = 3 (4) + 4 (5) $-$ 2
= 12 + 20 $-$ 2
= 30

#### Question 18:

If the binary operation * on the set Z of integers is defined by a * b = a + 3b2, find the value of 2 * 4.

Given: a * b = a + 3b2

Here,
2 * 4 = 2 + 3 (4)2
= 2 + 3 (16)
= 2 + 48
= 50

#### Question 19:

Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.

Given: a * b = HCF (a, b)

Here,

22 * 4 = HCF (22, 4)
= 2                                [because highest common factor of 22 and 4 is 2]

#### Question 20:

Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.

Given: a * b = 2a + b − 3

Here,

3 * 4 = 2 (3) + 4 $-$3
= 6 + 4 $-$ 3
= 7

#### Question 21:

If a * b denotes the larger of 'a' and 'b' and if aob = (a * b) + 3, then write the value of (5) o (10), where * and o are binary operations.

Given: a * b denotes the larger of 'a' and 'b'.
Also, $a\circ b=\left(a*b\right)+3$
For a = 5 and b = 10
a * b = 5 * 10 = 10
$a\circ b=5\circ 10=\left(5*10\right)+3=10+3=13$.

#### Question 1:

If a * b = a2 + b2, then the value of (4 * 5) * 3 is
(a) (42 + 52) + 32
(b) (4 + 5)2 + 32
(c) 412 + 32
(d) (4 + 5 + 3)2

(c) $\left({41}^{2}+{3}^{2}\right)$

Given: a * b = a2 + b2

#### Question 2:

If a * b denote the bigger among a and b and if ab = (a * b) + 3, then 4.7 =
(a) 14
(b) 31
(c) 10
(d) 8

(c) 10

4.7 = (4 * 7) + 3
= 7 + 3
=10

#### Question 3:

On the power set P of a non-empty set A, we define an operation ∆ by
Then which are of the following statements is true about ∆
(a) commutative and associative without an identity
(b) commutative but not associative with an identity
(c) associative but not commutative without an identity
(d) associative and commutative with an identity

(d) associative and commutative with an identity

Let $\varphi$ be the identity element for $∆$ on P.

#### Question 4:

If the binary operation * on Z is defined by a * b = a2b2 + ab + 4, then value of (2 * 3) * 4 is
(a) 233
(b) 33
(c) 55
(d) −55

(b) 33

Given: a * b = a2b2 + ab + 4

#### Question 5:

Mark the correct alternative in the following question:

For the binary operation * on Z defined by a * b = a + b + 1, the identity clement is

(a) 0                                   (b) $-$1                                   (c) 1                                   (d) 2

We have,

a * b = a + b + 1

Let e be the identity element of *. Then,

Hence, the correct alternative is option (b).

#### Question 6:

If a binary operation * is defined on the set Z of integers as a * b = 3ab, then the value of (2 * 3) * 4 is
(a) 2
(b) 3
(c) 4
(d) 5

(d) 5

Given: a * b = 3ab
2 * 3 = 3 (2) $-$ 3
= 6 $-$ 3
= 3

(2 * 3) * 4 = 3 * 4
= 3 (3) $-$ 4
= 9 $-$ 4
= 5

#### Question 7:

Q+ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q+ is defined as $a\odot =\frac{ab}{2}$, then the inverse of 3 is
(a) $\frac{4}{3}$

(b) 2

(c) $\frac{1}{3}$

(d) $\frac{2}{3}$

(a) $\frac{4}{3}$

Let e be the identity element in Q+ with respect to $\odot$ such that

Thus, 2 is the identity element in Q+ with respect to $\odot$.

#### Question 8:

If G is the set of all matrices of the form , then the identity element with respect to the multiplication of matrices as binary operation, is

(a) $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

(b) $\left[\begin{array}{cc}-1/2& -1/2\\ -1/2& -1/2\end{array}\right]$

(c) $\left[\begin{array}{cc}1/2& 1/1\\ 1/2& 1/2\end{array}\right]$

(d) $\left[\begin{array}{cc}-1& -1\\ -1& -1\end{array}\right]$

Disclaimer: The question in the book has some error, so, none of the options are matching with the solution. The solution is created according to the question given in the book.

#### Question 9:

Q+ is the set of all positive rational numbers with the binary operation * defined by for all a, bQ+. The inverse of an element aQ+ is
(a) a

(b) $\frac{1}{a}$

(c) $\frac{2}{a}$

(d) $\frac{4}{a}$

(d) $\frac{4}{a}$
Let e be the identity element in Q+ with respect to * such that

Thus, 2 is the identity element in Q+ with respect to *.

#### Question 10:

If the binary operation ⊙ is defined on the set Q+ of all positive rational numbers by is equal to
(a) $\frac{3}{160}$

(b) $\frac{5}{160}$

(c) $\frac{3}{10}$

(d) $\frac{3}{40}$

(a)  $\frac{3}{160}$

Given: $a\odot b=\frac{ab}{4}$

#### Question 11:

Let * be a binary operation defined on set Q − {1} by the rule a * b = a + bab. Then, the identify element for * is
(a) 1

(b) $\frac{a-1}{a}$

(c) $\frac{a}{a-1}$

(d) 0

(d) 0

Let e be the identity element in Q - {1} with respect to * such that

Thus, 0 is the identity element in Q $-$ {1} with respect to *.

#### Question 12:

Which of the following is true?
(a) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Z
(b) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Q
(c) all binary commutative operations are associative
(d) subtraction is a binary operation on N

(b) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Q.

Let us check each option one by one.

(a)

Hence, (a) is false.

(b)

Hence, (b) is true.

(c)
Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is not associative on N.
Therefore, all binary commutative operations are not associative.
Hence, (c) is false.

(d) Subtraction is not a binary operation on N because subtraction of any two natural numbers is not always a natural number.
For example: 2 and 4 are natural numbers.
2$-$4 = $-$2 which is not a natural number.
Hence, (d) is false.

#### Question 13:

The binary operation * defined on N by

a * b = a + b + ab for all a, bN is

(a) commutative only
(b) associative only
(c) commutative and associative both
(d) none of these

(c) commutative and associative both

Commutativity:

Thus, * is commutative on N.

Associativity:

Thus, * is associative on N.

#### Question 14:

The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to
(a) 20
(b) 40
(c) 400
(d) 445

(d) 445
Given: a * b = a2 + b2 + ab + 1

#### Question 15:

Let * be a binary operation on R defined by a * b = ab + 1. Then, * is
(a) commutative but not associative
(b) associative but not commutative
(c) neither commutative nor associative
(d) both commutative and associative

(a) commutative but not associative

Commutativity:

Therefore, * is commutative on R.

Associativity:

Hence, * is not associative on R.

#### Question 16:

Subtraction of integers is
(a) commutative but no associative
(b) commutative and associative
(c) associative but not commutative
(d) neither commutative nor associative

(d) neither commutative nor associative

Subtraction of integers is not commutative
For example: If a = 1 and b = 2, then both are integers

$⇒-1\ne 1$

Subtraction of integers is not associative.
For example: If a = 1, b = 2, c = 3, then all are integers

#### Question 17:

The law a + b = b + a is called
(a) closure law
(b) associative law
(c) commutative law
(d) distributive law

(c) commutative law

The law a + b = b + a is commutative.

#### Question 18:

An operation * is defined on the set Z of non-zero integers by $a*b=\frac{a}{b}$ for all a, bZ. Then the property satisfied is
(a) closure
(b) commutative
(c) associative
(d) none of these

(d) none of these

* is not closure because when a = 1 and b = 2,

* is not commutative because when a = 1 and b = 2,

* is not associative because when a = 1,  b = 2 and c = 3,

#### Question 19:

On Z an operation * is defined by a * b = a2 + b2 for all a, bZ. The operation * on Z is
(a) commutative and associative
(b) associative but not commutative
(c) not associative
(d) not a binary operation

(c) not associative

Commutativity:

Thus, * is commutative on Z.

Associativity:

Thus, * is not associative on Z.

#### Question 20:

A binary operation * on Z defined by a * b = 3a + b for all a, bZ, is
(a) commutative
(b) associative
(c) not commutative
(d) commutative and associative

(c) not commutative

Commutativity:

Thus, * is not commutative on Z.

#### Question 21:

Let * be a binary operation on Q+ defined by . The inverse of 0.1 is
(a) 105
(b) 104
(c) 106
(d) none of these

(a) 105

Let e be the identity element in Q+with respect to * such that

Thus, 100 is the identity element in Q+ with respect to *.

#### Question 22:

Let * be a binary operation on N defined by a * b = a + b + 10 for all a, bN. The identity element for * in N is
(a) −10
(b) 0
(c) 10
(d) non-existent

(d) non-existent

Let e be the identity element in N with respect to * such that

So, the identity element with respect to * does not exist in N.

#### Question 23:

Consider the binary operation * defined on Q − {1} by the rule
a * b = a + bab for all a, bQ − {1}
The identity element in Q − {1} is
(a) 0
(b) 1
(c) $\frac{1}{2}$
(d) −1

(a) 0

Let e be the identity element in Q $-$ {1} with respect to * such that

Thus, 0 is the identity element in Q $-$ {1} with respect to *.

#### Question 24:

For the binary operation * defined on R − {1} by the rule a * b = a + b + ab for all a, bR − {1}, the inverse of a is
(a) $-a$

(b) $-\frac{a}{a+1}$

(c) $\frac{1}{a}$

(d) ${a}^{2}$

(b)  $-\frac{a}{a+1}$

Let e be the identity element in R $-$ {1} with respect to * such that

Thus, 0 is the identity element in R $-$ {1}with respect to *.

#### Question 25:

For the multiplication of matrices as a binary operation on the set of all matrices of the form $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$, a, bR the inverse of $\left[\begin{array}{cc}2& 3\\ -3& 2\end{array}\right]$ is

(a) $\left[\begin{array}{cc}-2& 3\\ -3& -2\end{array}\right]$

(b) $\left[\begin{array}{cc}2& 3\\ -3& 2\end{array}\right]$

(c) $\left[\begin{array}{cc}2/13& -3/13\\ 3/13& 2/13\end{array}\right]$

(d) $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

(c) $\left[\begin{array}{cc}2/13& -3/13\\ 3/13& 2/13\end{array}\right]$

To find the identity element,

#### Question 26:

On the set Q+ of all positive rational numbers a binary operation * is defined by . The inverse of 8 is
(a) $\frac{1}{8}$

(b) $\frac{1}{2}$

(c) 2

(d) 4

(b) $\frac{1}{2}$

Let e be the identity element in Q+ with respect to * such that

Thus, 2 is the identity element in Q+ with respect to *.

#### Question 27:

Let * be a binary operation defined on Q+ by the rule . The inverse of 4 * 6 is

(a) $\frac{9}{8}$

(b) $\frac{2}{3}$

(c) $\frac{3}{2}$

(d) none of these

(a) $\frac{9}{8}$

Let e be the identity element in Q+ with respect to * such that

Thus, 3 is the identity element in Q+ with respect to *.

#### Question 28:

The number of binary operation that can be defined on a set of 2 elements is
(a) 8
(b) 4
(c) 16
(d) 64

(c) 16

We know that the number of binary operations on a set of n elements is ${n}^{{n}^{2}}$.

So, the number of binary operations on a set of 2 elements is

#### Question 29:

The number of commutative binary operations that can be defined on a set of 2 elements is
(a) 8
(b) 6
(c) 4
(d) 2

(d) 2
The number of commutative binary operations on a set of n elements is ${n}^{\frac{n\left(n-1\right)}{2}}\phantom{\rule{0ex}{0ex}}$.

Therefore,
Number of commutative binary operations on a set of 2 elements =

#### Question 1:

Determine whether each of the following operations define a binary operation on the given set or not :
(i)
(ii)
(iii) .

(iv)

(v)
(vi)
(vii)

Thus, * is a binary operation on N.

Thus, * is not a binary operation on  Z.

(iii)  If a = 1 and b = 1,
a * b = a + b$-$ 2
= 1 + 1$-$ 2
= 0
Thus, there exist a = 1 and b = 1 such that a * b
So, * is not a binary operation on N.

(iv) Consider the composition table,

 ${×}_{6}$ 1 2 3 4 5 1 1 2 3 4 5 2 2 4 0 2 4 3 3 0 3 0 3 4 4 2 0 4 2 5 5 4 3 2 1

Here all the elements of the table are not in S.

(v) Consider the composition table,

 ${+}_{6}$ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4

Here all the elements of the table are in S.

Thus, ${×}_{6}$ is a binary operation on S.

(vii) If a = 2 and b = $-$1 in Q,

So, * is not a binary operation on Q.

#### Question 2:

Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
(i) On Z+, defined * by a * b = ab
(ii) On Z+, defined * by a * b = ab
(iii) On R, define by a*b = ab2
(iv) On Z+ define * by a * b = |ab|
(v) On Z+, define * by a * b = a
(vi) On R, define * by a * b = a + 4b2
Here, Z+ denotes the set of all non-negative integers.

(i) If a = 1 and b = 2 in Z+, then

#### Question 3:

Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.

Given: a * b = 2a + b − 3

3 * 4 = 2 (3) + 4 $-$ 3
= 6 + 4 $-$ 3
= 7

#### Question 4:

Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.

 LCM 1 2 3 4 5 1 1 2 3 4 5 2 2 2 6 4 10 3 3 5 3 12 15 4 4 4 12 4 20 5 5 10 15 20 5

In the given composition table, all the elements are not in the set {1, 2, 3, 4, 5}.

If we consider a = 2 and b = 3, a * b = LCM of a and b = 6 $\notin${1, 2, 3, 4, 5}.

Thus, * is not a binary operation on {1, 2, 3, 4, 5}.

#### Question 5:

Let S = {a, b, c}. Find the total number of binary operations on S.

Number of binary operations on a set with n elements is ${n}^{{n}^{2}}$.

Here, S = {a, b, c}
Number of elements in S = 3
Number of binary operations on a set with 3 elements is ${3}^{{3}^{2}}={3}^{9}$

#### Question 6:

Find the total number of binary operations on {a, b}.

Number of binary operations on a set with n elements is ${n}^{{n}^{2}}$.

Here, S = {a, b}
Number of elements in S = 2

#### Question 7:

Let S be the set of all rational numbers of the form $\frac{m}{n}$, where m ∈ Z and n = 1, 2, 3. Prove that * on S defined by a * b = ab is not a binary operation.

Thus, * is not a binary operation.

#### Question 8:

Prove that the operation * on the set

defined by A * B = AB is a binary operation.

Thus, * is a binary operation on M.

#### Question 9:

The binary operation * : R $×$ R $\to$ R is defined as a * b = 2a + b. Find (2 * 3) * 4.                                                                     [CBSE 2012]

As, a * b = 2a + b

So, (2 * 3) * 4 = [2(2) + 3] * 4
= [4 + 3] * 4
= 7 * 4
= 2(7) + 4
= 14 + 4
= 18

#### Question 10:

Let * be a binary operation on N given by a * b = LCM (a, b) for all a, b $\in$ N. Find 5 * 7.                                                           [CBSE 2012]