1.

Determine whether or not 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 = 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 + 4b2Here, Z+ denotes the set of all non-negative integers.

Answer»

(i) Given that OnZ+, defined * by a * b = a – b

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

a * b = a – b

= 1 – 2

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

For a = 1 and b = 2,

a * b ∉ Z+

So, * is not a binary operation on Z+.

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

Let a, b ∈ Z+

⇒ a, b ∈ Z+

⇒ a * b ∈ Z+

So, * is a binary operation on R.

(iii) Given that on R, define by a*b = ab2

Let a, b ∈ R

⇒ a, b2 ∈ R

⇒ ab2 ∈ R

⇒ a * b ∈ R

So, * is a binary operation on R.

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

Let a, b ∈ Z+

⇒ |a – b| ∈ Z+

⇒ a * b ∈ Z+

So,

a * b ∈ Z+, ∀ a, b ∈ Z+

Thus, * is a binary operation on Z+.

(v) Given that on Z+ define * by a * b = a

Let a, b ∈ Z+

⇒ a ∈ Z+

⇒ a * b ∈ Z+

So, a * b ∈ Z+ ∀ a, b ∈ Z+

Thus, * is a binary operation on Z+.

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

Let a, b ∈ R

⇒ a, 4b2 ∈ R

⇒ a + 4b2 ∈ R

⇒ a * b ∈ R

So, a *b ∈ R, ∀ a, b ∈ R

Hence, * is a binary operation on R.


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