Final answer:
To show that the set P = a^2t^2 + a1t + a0 is a group under addition, we need to prove the closure property, associative property, commutative property, and identity property.
Step-by-step explanation:
To show that the set P = a2 + a1 = a0 and a2, a1, a0 ϵ R is a group under addition, we need to prove the closure property, associative property, commutative property, and identity property.
A) Closure Property:
To prove closure, we need to show that for any two elements in P, their sum is also in P. Let p1 = a1^2t^2 + a1t + a0 and p2 = a2^2t^2 + a2t + a0. The sum p1 + p2 = (a1^2 + a2^2)t^2 + (a1 + a2)t + 2a0 is also in P, since a1 + a2 = a0.
B) Associative Property:
To prove associativity, we need to show that for any three elements in P, their sum is independent of how we group them. Let p1, p2, and p3 be elements of P. (p1 + p2) + p3 = ((a1^2 + a2^2 + a3^2)t^2 + (a1 + a2 + a3)t + 3a0) + a4^2t^2 + a4t + a0, which is equivalent to (a1^2 + a2^2 + a3^2 + a4^2)t^2 + (a1 + a2 + a3 + a4)t + (4a0 + a0). Similarly, p1 + (p2 + p3) leads to the same result, proving associativity.
C) Commutative Property:
The commutative property states that the order of addition does not matter. Let p1 and p2 be elements of P. p1 + p2 = (a1 + a2)t^2 + (a1 + a2)t + 2a0. Similarly, p2 + p1 = (a2 + a1)t^2 + (a2 + a1)t + 2a0. Since addition is commutative, p1 + p2 = p2 + p1.
D) Identity Property:
The identity element is the element that, when added to any element, gives back the same element. In this case, the identity element is the element with a2 = a1 = a0 = 0. Adding any element p = a^2t^2 + a1t + a0 to the identity element gives p, satisfying the identity property.