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QUESTION: 1

Number of binary sets on the set {p, q,r} is:

Solution:

**Correct Answer :- a**

**Explanation : **No. of elements : 3

Total no.of binary operations on a set consisting 'n' elements is given by n^{n^2}

= So the total no.of binary sets is 3^{9}.

QUESTION: 2

* is said to be commutative in A for all a,b ε A

Solution:

QUESTION: 3

Given below is the table corresponding to some binary operation a * b on a set {0,1,2,3,4,5}. How many elements of this operation have an inverse?.

Solution:

QUESTION: 4

If * and ^{O} are two binary operations defined by a * b = a + b and a ^{O} b = ab, then

Solution:

QUESTION: 5

The law a + b = b + a is called

Solution:

Commutative law, in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + b = b + a and ab = ba

QUESTION: 6

The number of binary operations which can be defined on the set P= {p, q} is

Solution:

n(A) = n^{n^2}

=2^{2^2}

= 2^{4}

= 16

QUESTION: 7

Given below is binary composition table a* b = LCM of a and b on S = {1,2,3,4}. Then, from the table determine which one of these options is correct.

Solution:

We have ∗ is defined on the set {1,2,3,4,5} by a∗b = LCM of a and b

Now, 3∗4 = LCM of 3 and 4=12

But, 12 does not belong to {1,2,3,4,5}.

Hence, ′∗′ is not a binary operation.

QUESTION: 8

Let * be any binary operation on the set R defined by a * b = a + b – ab, then the binary operation * is

Solution:

Given that ο is a binary operation on Q – { – 1} defined by aοb = a + b – ab for all a,b∈Q – { – 1}.

We know that commutative property is pοq = qοp, where ο is a binary operation.

Let’s check the commutativity of given binary operation:

⇒ aοb = a + b – ab

⇒ bοa = b + a – ba = a + b – ab

⇒ b*a = a*b

∴ Commutative property holds for given binary operation ‘ο’ on ‘Q – [ – 1]’.

We know that associative property is (pοq)οr = pο(qοr)

Let’s check the associativity of given binary operation:

⇒ (aοb)οc = (a + b – ab)οc

⇒ (aοb)οc = a + b – ab + c – ((a + b – ab)×c)

⇒ (aοb)οc = a + b + c – ab – ac – ab + abc ...... (1)

⇒ aο(bοc) = aο(b + c – bc)

⇒ aο(bοc) = a + b + c – bc – (a×(b + c – bc))

⇒ a*(b*c) = a + b + c – ab – bc – ac + abc ...... (2)

From (1) and (2)

⇒ (aob)oc = ao(boc)

Hence,we can clearly say that associativity hold for the given binary operation on ‘Q – { – 1}’.

Given that ο is a binary operation on Q – { – 1} defined by aοb = a + b – ab for all a,b∈Q – { – 1}.

We know that commutative property is pοq = qοp, where ο is a binary operation.

Let’s check the commutativity of given binary operation:

⇒ aοb = a + b – ab

⇒ bοa = b + a – ba = a + b – ab

⇒ b*a = a*b

∴ Commutative property holds for given binary operation ‘ο’ on ‘Q – [ – 1]’.

We know that associative property is (pοq)οr = pο(qοr)

Let’s check the associativity of given binary operation:

⇒ (aοb)οc = (a + b – ab)οc

⇒ (aοb)οc = a + b – ab + c – ((a + b – ab)×c)

⇒ (aοb)οc = a + b + c – ab – ac – ab + abc ...... (1)

⇒ aο(bοc) = aο(b + c – bc)

⇒ aο(bοc) = a + b + c – bc – (a×(b + c – bc))

⇒ a*(b*c) = a + b + c – ab – bc – ac + abc ...... (2)

From (1) and (2)

⇒ (aob)oc = ao(boc)

Hence,we can clearly say that associativity hold for the given binary operation on ‘Q – { – 1}’.

QUESTION: 9

Given below is the table corresponding to some binary operation a * b on a set {0, 1,2,3,4,5}. Identify the identity element of this operation.

Solution:

QUESTION: 10

The identity element for the binary operation * defined by a * b = ab/2, where a, b are the elements of a set of non-zero rational numbers, is

Solution:

a*e = a = e*a

ae/2 = a

e = 2

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