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Question Number 190241 by astridmei last updated on 30/Mar/23
show that a⊛b=a+ab+b is a monoid when G=Z
showthata⊛b=a+ab+bisamonoidwhenG=Z
Answered by mehdee42 last updated on 30/Mar/23
∗:Z×Z→Z ; a∗b=a+b+ab (Z,∗) is monoid whenever 1)∀a,b∈Z →a∗b∈Z 2)∀a,b,c∈Z→(a∗b)∗c=a∗(b∗c) 3)∃e∈Z ;∀a∈Z →a∗ e=e∗a=a ∴ 1)a∗b=a+b+ab∈Z ✓ 2)(a∗b)∗c=(a+b+ab)∗c=a+b+ab+c+(a+b+ab)c=a+b+c+ab+ac+bc+abc a∗(b∗c)=a∗(b+c+bc)=a+b+c+bc+a(b+c+bc)=a+b+c+ab+ac+bc+abc ⇒(2)✓ 3) ∀ a∈Z →a∗0=o∗a=a ✓
∗:Z×Z→Z;a∗b=a+b+ab(Z,∗)ismonoidwhenever1)∀a,b∈Z→a∗b∈Z2)∀a,b,c∈Z→(a∗b)∗c=a∗(b∗c)3)∃e∈Z;∀a∈Z→a∗e=e∗a=a∴1)a∗b=a+b+ab∈Z✓2)(a∗b)∗c=(a+b+ab)∗c=a+b+ab+c+(a+b+ab)c=a+b+c+ab+ac+bc+abca∗(b∗c)=a∗(b+c+bc)=a+b+c+bc+a(b+c+bc)=a+b+c+ab+ac+bc+abc⇒(2)✓3)∀a∈Z→a∗0=o∗a=a✓

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