Question

Exorcuses for Lede s IPy 47 yiven a set V of variables, let E, be o et of expre ssions de fined as follows: El and Ea belong to Ev , then CEbelangs to ey (he purns İSinno eiTher E is a varable belonging to V,or has the form (x+Ewheu oom Prove:f is asseciative, then every E e Ey can be trunsfprme d nto normal rm

Answer 1

Let E be an expression in E_v.

Observe that we can write this expression as follows:

E=(((...(x+E₁)+E₂)...)+E_n by the construction of the set E_v. where x is a variable, since to form any expression we must first start with a variable from the set V.

Now if + is associative, the expression can be written as:

E=x+((..(E₁+E₂)+..)+E_n)

Let E^'=((..(E₁+E₂)+..)+E_n) (clearly this is in E_v)

Then E=x+E^'

Thus transforming E into normal form.