Let * be a binary operation on the set of real numbers R defined as follows: a*b=a+b-3(ab)^2

Brock Brown 2022-01-07 Answered
Let * be a binary operation on the set of real numbers R defined as follows:
ab=a+b3(ab)2, where a,bR
- Prove that * is commutative but not associative algebraic operation on R.
- Find the identity element for * .
- Show that 1 has two inverses with respect to *.
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Expert Answer

Donald Cheek
Answered 2022-01-08 Author has 41 answers
Step 1
The statement is commutative if ab=ba and associative if (ab)c=a(bc) Step 2
Calculate ab and ba
ab=a+b3(ab)2
ba=b+a3(ba)2
=a+b3(ab)2
=ab
Step 3
Calculate (ab)c and a(bc).
(ab)c=(a+b3(ab)2c
=(a+b3(ab)2)+c3[(a+b3(ab)2)c]2
=a+b+c3a2b23c2(a+b3(ab)2)2
a(bc)=a(b+c3(bc)2)
=a+(b+c3(bc)2)3[a(b+c3(bc)2)]2
=a+b+c3b2c23a2(b+c3(bc2))2
(ab)c
Thus, not associative.
Step 5
Determine identity.
ae=a
a+e3(ae)2=a
e3e2a2=0
e(13e2a2)=0
e=0,13e2a2=0
e=0,e2=13a2
e=0,e=13a
Step 6
For inverse a1 following condition needs to be satisfied.

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