Answer to Question #123753 in Abstract Algebra for Nii Laryea

Question

Show that the set P = {a₂t² + a₁t + a₀ |a₂+ a₁ = a₀ and a₂, a₁, a₀ ϵ R} is a group under addition

Answer 1

Given $P = \{a_2 t² + a_1 t + a_0 | a_2+ a_1 = a_0 \ and \ a_2, a_1, a_0 \in \R\}$ .

(i) Closure: Let $A\in P, B \in P$ , so $A = a_2 t² + a_1 t + a_0 , B = a_2' t² + a_1' t + a_0'$

and $a_2+a_1=a_0, a_2'+a_1'=a_0'$ .

So, $(a_2+a_2')+(a_1+a_1') = a_0 +a_0'$ and $A+B \in P$ . Hence, Closure property holds.

(ii) Also, $(A+B)+C=A+(B+C) \ \forall \ A,B,C \in P$ .

Hence, Associativity property holds,

(iii) Identity: $\exists \ 0\in P : 0+A = A$ for all $A \in P$ . Hence, 0 is identity exists in P.

(iv) Inverse: $\forall \ A\in P$ , $\exist \ B=-A\in P : A+B=0$ .

Hence, the set P is group under addition.