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# Defined a binary operation ∗ on the set {0,1,2,3,4,5} asa∗b={a+b,if a+b<6a+b−6if a+b≥6Show that zero is the identity for this operation and each element a≠0 of the set is invertible with 6−a being the inverse of a.

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Solution

## 0∗0=0+0=00∗1=0+1=10∗2=0+2=20∗3=0+3=30∗4=0+4=40∗5=0+5=5i.e 0∗a=0+a=aand a∗0=a+0=a ⇒a∗0=0∗a=a ∀a∈{ 0,1,2,3,4,5} ⇒ 0 is the identity for this operation Let a∈{ 0,1,2,3,4,5} and b=6−aThen a∗b=a+b−6 =a+6−a−6 =0 (since a+b=6)and b∗a=b+a−6 =6−a+a−6 (since a+b=6)⇒Each element a∈{ 0,1,2,3,4,5} is invertible and has 6−a as its inverse

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