9 of 10 (8 complete) Decide which properties of the single-operation system are

Question

9 of 10 (8 complete) Decide which properties of the single-operation system are satisfied. If the identity property is satisfied, give the identity element. If the inverse.property is satisfied, give the inverse of each element. is the closure property satisfied for the system? O No Yes is the commutative property satisfied for the system? O No Yes is the associative property satisfied for the system? O Yes O No is the identity property satisfied for the system? O Yes No What is the identity element? Select the correct choice below and fill in any answer boxes within your choice. O A. The identity of the element is O B. There is no identity element Click to select and enter your answer(s).

Explanation / Answer

The elements of given set are S = { C, D, H}

(a) Closure property is satisfied because all the elements of the table are present in S.

So answer is "yes"

(b) Commutative property is satisfied because the matrix formed by the rows and columns of the table is symmetric.
So the answer is "yes"

(c) Associative property is satisfied because,
(C x D)x H = HxH = H
C x (DxH) = Cx D = H
So, (C x D)x H =C x (DxH).
So, the answer is "yes".

(d) Identity element is H because the row corresponding to H is just same as the top most row.
So, the identity element is "H".

(e) The inverse property is satisfied because, each row corresponding to C, D and H has the identity element H in it.

It means,

C x D = H. So, inverse of C = D

D x C = H. So inverse of D = C

H x H = H. So inverse of H = H.

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