Please take a look at this problem.

1. Determine whether or not, each of the following sets is closed under the given operations defined by:

[imath]\qquad \textrm{(a) } a * b = 4(a + b),\; a,\,b \in \mathbb{R}[/imath]

[imath]\qquad \textrm{(b) } p \nabla q = \dfrac{pq}{5},\; p,\,q \in \mathbb{R}[/imath]

[imath]\qquad \textrm{(c) } x \Omicron y = x + y + \dfrac{xy}{3},\; x,\,y \in \mathbb{Q}[/imath]

[imath]\qquad \textrm{(d) } a \nabla b = \vert a - b \vert,\; a,\,b \in\mathbb{N}[/imath]

I decided to go for (d):

Suppose a=1,b=2 then

[imath]\qquad 1-2=-1[/imath]

The set N is not closed under the given binary operations because:

[imath]\qquad -1 \cancel{\epsilon} N[/imath]

But on the other hand, if a=2, b=1 then:

[imath]\qquad 2-1=1[/imath]

I can now say that the set N, is closed because:

[imath]\qquad 1 \in N[/imath]

What do you have to say about my idea on this problem?

1. Determine whether or not, each of the following sets is closed under the given operations defined by:

[imath]\qquad \textrm{(a) } a * b = 4(a + b),\; a,\,b \in \mathbb{R}[/imath]

[imath]\qquad \textrm{(b) } p \nabla q = \dfrac{pq}{5},\; p,\,q \in \mathbb{R}[/imath]

[imath]\qquad \textrm{(c) } x \Omicron y = x + y + \dfrac{xy}{3},\; x,\,y \in \mathbb{Q}[/imath]

[imath]\qquad \textrm{(d) } a \nabla b = \vert a - b \vert,\; a,\,b \in\mathbb{N}[/imath]

I decided to go for (d):

Suppose a=1,b=2 then

[imath]\qquad 1-2=-1[/imath]

The set N is not closed under the given binary operations because:

[imath]\qquad -1 \cancel{\epsilon} N[/imath]

But on the other hand, if a=2, b=1 then:

[imath]\qquad 2-1=1[/imath]

I can now say that the set N, is closed because:

[imath]\qquad 1 \in N[/imath]

What do you have to say about my idea on this problem?

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