MENU
Question -

Determine whether or not the definition of * given below gives a binaryoperation. In the event that * is not a binary operation give justification ofthis.
(i) On Z+, defined *by a * b = a – b

(ii) On Z+, define * by a*b = ab

(iii) On R, define * by a*b = ab2

(iv) On Z+ define * by a * b =|a − b|

(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-negativeintegers.



Answer -

(i) Given On Z+,defined * by a * b = a – b

If a = 1 and b = 2 inZ+, then

a * b = a – b

= 1 – 2

= -1 ∉ Z+ [because Z+ isthe set of non-negative integers]

For a = 1 and b = 2,

a * b ∉ Z+

Thus, * is not abinary operation on Z+.

(ii) Given Z+,define * by a*b = a b

Let a, b ∈ Z+

⇒ a, b ∈ Z+

⇒ a * b ∈ Z+

Thus, * is a binaryoperation on R.

(iii) Given on R,define by a*b = ab2

Let a, b ∈ R

⇒ a, b2 ∈ R

⇒ ab2 ∈ R

⇒ a * b ∈ R

Thus, * is a binaryoperation on R.

(iv) Given on Z+ define* by a * b = |a − b|

Let a, b ∈ Z+

⇒ | a – b | ∈ Z+

⇒ a * b ∈ Z+

Therefore,

a * b ∈ Z+, ∀ a, b ∈ Z+

Thus, * is a binaryoperation on Z+.

(v) Given on Z+ define* by a * b = a

Let a, b ∈ Z+

⇒ a ∈ Z+

⇒ a * b ∈ Z+

Therefore, a * b ∈ Z+ ∀ a, b ∈ Z+

Thus, * is a binaryoperation on Z+.

(vi) Given On R,define * by a * b = a + 4b2

Let a, b ∈ R

⇒ a, 4b2 ∈ R

⇒ a + 4b2 ∈ R

⇒ a * b ∈ R

Therefore, a *b ∈ R, ∀ a, b ∈ R

Thus, * is a binaryoperation on R.

Comment(S)

Show all Coment

Leave a Comment

Free - Previous Years Question Papers
Any questions? Ask us!
NodeJS Interview Questions Answers
ASP.Net Interview Questions Answers
Interview Questions Answers
×