Question

i have couple proof problems with a field of given axioms, please help

Answer 1

axioms:
i) ∀a,b,c∈F, (a+b)+c=a+(b+c)
ii) ∀a∈F, 0+a=a
iii) ∀a∈F, ∃a ̃∈F, a ̃+a=0
iv) ∀a,b∈F, a+b=b+a
v) ∀a,b,c ∈ F, (ab)c = a(bc)
vi) ∀a∈F, 1a=a
vii) ∀a∈F, ∼(a=0)⇒∃a′ ∈F,a′a=1
viii) ∀a, b ∈ F, ab = ba
ix) ∀a,b,c∈F, (a+b)c=ac+bc

problem
3) If 0 = 1, then ∀a ∈ F,a = 0.
(5) Ifa∈F and x∈F is an element such that x+a=0,then x =a'
(6) ∀a∈A,a''=a
(7) ∀a∈1'a=a'

(a' inverse)