**Question 1: Determine whether or not each of the definition of âˆ— given below gives a binary operation. In the event that âˆ— is not a binary operation, give justification for this.**

**(i) On Z+, define âˆ— by a âˆ— b = a â€“ b**

**Solution: **

If a, b belongs to Z+

a * b = a – b which may not belong to Z+

For eg: 1 – 3 = -2 which doesn’t belongs to Z+

Therefore, * is not a Binary Operation on Z+

**(ii) On Z+, define * by a * b = ab**

**Solution: **

If a, b belongs to Z+

a * b = ab which belongs to Z+

Therefore, * is Binary Operation on Z+

**(iii) On R, define * by a * b = abÂ²**

**Solution:**

If a, b belongs to R

a * b = ab

^{2 }which belongs to RTherefore, * is Binary Operation on R

**(iv) On Z+, define * by a * b = |a â€“ b|**

**Solution:**

If a, b belongs to Z+

a * b = |a – b| which belongs to Z+

Therefore, * is Binary Operation on Z+

**(v) On Z+, define * by a * b = a**

**Solution:**

If a, b belongs to Z+

a * b = a which belongs to Z+

Therefore, * is Binary Operation on Z+

**Question 2: For each binary operation * defined below, determine whether * is binary, commutative or associative.**

**(i) On Z, define a * b = a â€“ b **

**Solution:**

a) Binary:If a, b belongs to Z

a * b = a – b which belongs to Z

Therefore, * is Binary Operation on Z

b) Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = a – b

RHS = b * a = b – a

Since, LHS is not equal to RHS

Therefore, * is not Commutative

c) Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a – b + c

RHS = (a – b) * c = a – b- c

Since, LHS is not equal to RHS

Therefore, * is not Associative

**(ii) On Q, define a * b = ab + 1**

**Solution:**

a) Binary:If a, b belongs to Q, a * b = ab + 1 which belongs to Q

Therefore, * is Binary Operation on Q

b) Commutative:If a, b belongs to Q, a * b = b * a

LHS = a * b = ab + 1

RHS = b * a = ba + 1 = ab + 1

Since, LHS is equal to RHS

Therefore, * is Commutative

c) Associative:If a, b, c belongs to Q, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * (bc + 1) = abc + a + 1

RHS = (a * b) * c = abc + c + 1

Since, LHS is not equal to RHS

Therefore, * is not Associative

**(iii) On Q, define a âˆ— b = ab/2**

**Solution :**

a) Binary:If a, b belongs to Q, a * b = ab/2 which belongs to Q

Therefore, * is Binary Operation on Q

b) Commutative:If a, b belongs to Q, a * b = b * a

LHS = a * b = ab/2

RHS = b * a = ba/2

Since, LHS is equal to RHS

Therefore, * is Commutative

c) Associative:If a, b, c belongs to Q, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * (bc/2) = (abc)/2

RHS = (a * b) * c = (ab/2) * c = (abc)/2

Since, LHS is equal to RHS

Therefore, * is Associative

**(iv) On Z+, define a * b = 2**^{ab}

^{ab}

**Solution:**

a) Binary:If a, b belongs to Z+, a * b = 2

^{ab}which belongs to Z+Therefore, * is Binary Operation on Z+

b) Commutative:If a, b belongs to Z+, a * b = b * a

LHS = a * b = 2

^{ab}RHS = b * a = 2

^{ba}= 2^{ab}Since, LHS is equal to RHS

Therefore, * is Commutative

c) Associative:If a, b, c belongs to Z+, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * 2

^{bc = }2^{a * 2^(bc)}RHS = (a * b) * c = 2

^{ab}* c = 2^{2abc}Since, LHS is not equal to RHS

Therefore, * is not Associative

**(v) On Z+, define a * b = a**^{b}

^{b}

**Solution:**

a) Binary:If a, b belongs to Z+, a * b = a

^{b}which belongs to Z+Therefore, * is Binary Operation on Z+

b) Commutative:If a, b belongs to Z+, a * b = b * a

LHS = a * b = a

^{b}RHS = b * a = b

^{a}Since, LHS is not equal to RHS

Therefore, * is not Commutative

c) Associative:If a, b, c belongs to Z+, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * b

^{c}= a^{b^c}RHS = (a * b) * c = a

^{b}* c = a^{bc}Since, LHS is not equal to RHS

Therefore, * is not Associative

**(vi) On R â€“ {â€“ 1}, define a âˆ— b = a / (b + 1)**

**Solution: **

a) Binary:If a, b belongs to R, a * b = a / (b+1) which belongs to R

Therefore, * is Binary Operation on R

b) Commutative:If a, b belongs to R, a * b = b * a

LHS = a * b = a / (b + 1)

RHS = b * a = b / (a + 1)

Since, LHS is not equal to RHS

Therefore, * is not Commutative

c) Associative:If a, b, c belongs to A, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * b / (c+1) = a(c+1) / b+c+1

RHS = (a * b) * c = (a / (b+1)) * c = a / (b+1)(c+1)

Since, LHS is not equal to RHS

Therefore, * is not Associative

**Question 3. Consider the binary operation âˆ§ on the set {1, 2, 3, 4, 5} defined by a âˆ§ b = min {a, b}. Write the operation table of the operation âˆ§. **

**Solution: **

^ 1 2 3 4 5 1 1 1 1 1 1 2 1 2 2 2 2 3 1 2 3 3 3 4 1 2 3 4 4 5 1 2 3 4 5

**Question 4: Consider a binary operation âˆ— on the set {1, 2, 3, 4, 5} given by the following multiplication table.**

**(Hint: use the following table) **

* |
1 |
2 |
3 |
4 |
5 |

1 |
1 |
1 |
1 |
1 |
1 |

2 |
1 |
2 |
1 |
2 |
1 |

3 |
1 |
1 |
3 |
1 |
1 |

4 |
1 |
2 |
1 |
4 |
1 |

5 |
1 |
1 |
1 |
1 |
5 |

**(i) Compute (2 âˆ— 3) âˆ— 4 and 2 âˆ— (3 âˆ— 4)**

**Solution:**

Here, (2 * 3) * 4 = 1 * 4 = 1

2 * (3 * 4) = 2 * 1 = 1

**(ii) Is âˆ— commutative?**

**Solution:**

The given composition table is symmetrical about the main diagonal of table. Thus, binary operation ‘*’ is commutative.

**(iii) Compute (2 âˆ— 3) âˆ— (4 âˆ— 5).**

**Solution:**

(2 * 3) * (4 * 5) = 1 * 1 = 1

**Question 5: Let âˆ—â€² be the binary operation on the set {1, 2, 3, 4, 5} defined by a âˆ—â€² b = H.C.F. of a and b. Is the operation âˆ—â€² same as the operation âˆ— defined in Exercise 4 above? Justify your answer.**

**Solution:**

Let A = {1, 2, 3, 4, 5} and a âˆ—â€² b = HCF of a and b.

*’ 1 2 3 4 5 1 1 1 1 1 1 2 1 2 1 2 1 3 1 1 3 1 1 4 1 2 1 4 1 5 1 1 1 1 5 We see that the operation *’ is the same as the operation * in Exercise 4 above.

**Question 6: Let âˆ— be the binary operation on N given by a âˆ— b = L.C.M. of a and b. Find**

**(i) 5 âˆ— 7, 20 âˆ— 16**

**Solution:**

If a, b belongs to N

a * b = LCM of a and b

5 * 7 = 35

20 * 16 = 80

**(ii) Is âˆ— commutative?**

**Solution:**

If a, b belongs to N

LCM of a * b = ab

LCM of b * a = ba = ab

a*b = b*a

Thus, * binary operation is commutative.

**(iii) Is âˆ— associative?**

**Solution:**

a * (b * c) = LCM of a, b, c

(a * b) * c = LCM of a, b, c

Since, a * (b * c) = (a * b) * c

Thus, * binary operation is associative.

**(iv) Find the identity of âˆ— in N**

**Solution:**

Let ‘e’ is an identity

a * e = e * a, for a belonging to N

LCM of a * e = a, for a belonging to N

LCM of e * a = a, for a belonging to N

e divides a

e divides 1

Thus, e = 1

Hence, 1 is an identity element

**(v) Which elements of N are invertible for the operation âˆ—? **

**Solution:**

a * b = b * a = identity element

LCM of a and b = 1

a = b = 1

only ‘1’ is invertible element in N.

Question 7: Is âˆ— defined on the set {1, 2, 3, 4, 5} by a âˆ— b = L.C.M. of a and b a binary operation? Justify your answer.

Solution:The operation * on the set {1, 2, 3, 4, 5} is defined as

a * b = L.C.M. of a and b

Let a=3, b=5

3 * 5 = 5 * 3 = L.C.M. of 3 and 5 = 15 which does not belong to the given set

Thus, * is not a Binary Operation.

Question 8: Let âˆ— be the binary operation on N defined by a âˆ— b = H.C.F. of a and b. Is âˆ— commutative? Is âˆ— associative? Does there exist identity for this binary operation on N?

Solution:If a, b belongs to N

LHS = a * b = HCF of a and b

RHS = b * a = HCF of b and a

Since LHS = RHS

Therefore, * is Commutative

Now, If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = HCF of a, b and c

RHS = (a – b) * c = HCF of a, b and c

Since, LHS = RHS

Therefore, * is Associative

Now, 1 * a = a * 1 â‰ a

Thus, there doesn’t exist any identity element.

Question 9: Let âˆ— be a binary operation on the set Q of rational numbers as follows:

(i) a âˆ— b = a â€“ b

(ii) a âˆ— b = a^{2}+ b^{2}

(iii) a âˆ— b = a + ab

(iv) a âˆ— b = (a â€“ b)^{2}

(v) a âˆ— b = ab / 4

(vi) a âˆ— b = ab^{2}

Find which of the binary operations are commutative and which are associative.

Solution:

(i) Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = a – b

RHS = b * a = b – a

Since, LHS is not equal to RHS

Therefore, * is not Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a – (b – c) = a – b + c

RHS = (a – b) * c = a – b – c

Since, LHS is not equal to RHS

Therefore, * is not Associative

(ii) Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = a

^{2}+ b^{2}RHS = b * a = b

^{2}+ a^{2}Since, LHS is equal to RHS

Therefore, * is Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * (b

^{2 }+ c^{2}) = a^{2 }+ (b^{2}+ c^{2})^{2}RHS = (a * b) * c = (a

^{2}+ b^{2}) * c = (a^{2}+ b^{2})^{2}+ c^{2}Since, LHS is not equal to RHS

Therefore, * is not Associative

(iii) Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = a + ab

RHS = b * a = b + ba

Since, LHS is not equal to RHS

Therefore, * is not Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * (b + bc) = a + a(b + bc)

RHS = (a * b) * c = (a + ab) * c = a + ab + (a + ab)c

Since, LHS is not equal to RHS

Therefore, * is not Associative

(iv) Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = (a – b)

^{2}RHS = b * a = (b – a)

^{2}Since, LHS is not equal to RHS

Therefore, * is not Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * (b – c)

^{2}= [a – (b – c)^{2}]^{2}RHS = (a * b) * c = (a – b)

^{2}* c = [(a – b)^{2}– c]^{2}Since, LHS is not equal to RHS

Therefore, * is not Associative

(v) Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = ab / 4

RHS = b * a = ba / 4

Since, LHS is equal to RHS

Therefore, * is Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * bc/4 = abc/16

RHS = (a * b) * c = ab/4 * c = abc/16

Since, LHS is equal to RHS

Therefore, * is Associative

(vi) Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = ab

^{2}RHS = b * a = ba

^{2}Since, LHS is not equal to RHS

Therefore, * is not Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * (bc)

^{2}= a(bc^{2})^{2}RHS = (a * b) * c = (ab

^{2}) * c = ab^{2}c^{2}Since, LHS is not equal to RHS

Therefore, * is not Associative

Question 10: Find which of the operations given above has identity

Solution:An element e âˆˆ Q will be the identity element for the operation * if

a * e = a = e * a, for a âˆˆ Q

for (v) a * b = ab/4

Let e be an identity element

a * e = a = e * a

LHS : ae/4 = a

=> e = 4

RHS : ea/4 = a

=> e = 4

LHS = RHS

Thus, Identity element exists

Other operations doesn’t satisfy the required conditions.

Hence, other operations doesn’t have identity.

Question 11: Let A = N Ã— N and âˆ— be the binary operation on A defined by :

(a, b) âˆ— (c, d) = (a + c, b + d)

Show that âˆ— is commutative and associative. Find the identity element for âˆ— on A, if any.

Solution:Given (a, b) * (c, d) = (a+c, b+d) on A

Let (a, b), (c, d), (e,f) be 3 pairs âˆˆ A

Commutative :LHS = (a, b) * (c, d) = (a+c, b+d)

RHS = (c, d) * (a, b) = (c+a, d+b) = (a+c, b+d)

Since, LHS is equal to RHS

Therefore, * is Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = (a, b) * [(c, d) * (e, f)] = (a, b) * (c+e, d+f) = (a+c+e, b+d+f)

RHS = [(a, b) * (c, d)] * (e, f) = (a+c, b+d) * (e, f) = (a+c+e, b+d+f)

Since, LHS is equal to RHS

Therefore, * is Associative

Existence of Identity element:For a, e âˆˆ A, a * e = a

(a, b) * (e, e) = (a, b)

(a+e, b+e) = (a, b)

a + e = a

=> e = 0

b + e = b

=> e = 0

As 0 is not a part of set of natural numbers. So, identity function does not exist.

Question 12: State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation âˆ— on a set N, a âˆ— a = a âˆ€ a âˆˆ N.

(ii) If âˆ— is a commutative binary operation on N, then a âˆ— (b âˆ— c) = (c âˆ— b) âˆ— a

Solution:

(i)Let * be an operation on N, defined as:a * b = a + b âˆ€ a, b âˆˆ N

Let us consider b = a = 6, we have:

6 * 6 = 6 + 6 = 12 â‰ 6

Therefore, this statement is false.

(ii)Since, * is commutativeLHS = a âˆ— (b âˆ— c) = a * (c * b) = (c * b) * a = RHS

Therefore, this statement is true.

Question 13: Consider a binary operation âˆ— on N defined as a âˆ— b = a^{3}+ b^{3}. Choose the correct answer.

(A) Is âˆ— both associative and commutative?

(B) Is âˆ— commutative but not associative?

(C) Is âˆ— associative but not commutative?

(D) Is âˆ— neither commutative nor associative?

Solution:On N, * is defined as a * b = a

^{3}+ b^{3}

Commutative:If a, b belongs to Z, a * b = b * a

LHS = a * b = a

^{3}+ b^{3}RHS = b * a = b

^{3}+ a^{3}Since, LHS is equal to RHS

Therefore, * is Commutative

Associative:If a, b, c belongs to Z, a * (b * c) = (a * b) * c

LHS = a * (b * c) = a * (b

^{3}+ c^{3}) = a^{3}+ (b^{3}+ c^{3})^{3}RHS = (a * b) * c = (a

^{3}+ b^{3}) * c = (a^{3}+ b^{3})^{3}+ c^{3}Since, LHS is not equal to RHS

Therefore, * is not Associative

Thus, Option (B) is correct.

Share your thoughts in the comments## Please

Loginto comment...