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Question

Let ' * ' be a binary operation on set Q {1} defined bya * b = a + b ab for all a, b Q {1}.
Then, which of the following statement(s) is/are true?

A
0 is the identity element with respect to * on Q{1}.
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B
Every element of Q {1} is invertible.
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C
For any element aQ{1}, inverse of a is aa1
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D
* is associative on Q {1}
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Solution

The correct options are
A 0 is the identity element with respect to * on Q{1}.
B Every element of Q {1} is invertible.
C For any element aQ{1}, inverse of a is aa1
D * is associative on Q {1}
Let e be the identity element.a*e=e*a=a aQ(1}a+eae=aeae=0e(1a)=0e=0 (Since a1] The identity element with respect to '*' on Q(1} is 0.If 'b' is the inverse element of 'a' wrt * on Q(1}, thena*b=b*a=e, aQ(1}a+bab=0 a+b(1a)=0b=aa1Here, b is defined for all aQ(1}.Every element of Q(1} is invertible and the inverse is given by aa1.

Let a, b, cQ(1}Consider (a*b)*c =(a+bab)*c =a+bab+c(a+bab)c =a+b+cabbccaabcNow, a*(b*c) =a*(b+cbc) =a+b+cbca(b+cbc) =a+b+cabbccaabcThus, (a*b)*c= a*(b*c)for all a, b, cQ(1} * is associative on Q(1}.

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