More precisely, the power set 2 Θ is defined as the set of all composite propositions/subsets built from elements of Θ with ∪ operator such that:
1. ∅, θ1**, . . . , θ**n ∈ 2 Θ.

2. If **A, B** ∈ 2 Θ, then A ∪ B ∈ 2 Θ.

3. No other elements belong to 2 Θ, except those obtained by using rules 1 and 2.

## Examples Of Power Sets:

- If Θ = {θ1, θ2}, then 2 Θ={θ1,θ2} = {{∅}, {θ1}, {θ2}, {θ1, θ2}} which is commonly written as 2 Θ =
{∅, θ1, θ2, θ1 ∪ θ2}.

- Let's consider two frames Θ1 = {**A, B**} and Θ2 = {**X, Y** }, then their power sets are respectively 2 Θ1={A,B} = {∅**, A, B, A** ∪ B} and 2 Θ2={X,Y } = {∅**, X, Y, X** ∪ Y }. Let's consider a refined frame Θref = {θ1, θ2, θ3, θ4}. The granules θi, i = 1**, . . . ,** 4 are not necessarily exhaustive, nor exclusive. If A and B are expressed more precisely in function of the granules θi by example as A , {θ1, θ2, θ3} ≡ θ1 ∪ θ2 ∪ θ3 and B , {θ2, θ4} ≡ θ2 ∪ θ4 then the power sets can be expressed from the granules θi as follows:
$$2^{\Theta_{1}=\{A,B\}}=\{\emptyset,A,B,A\cup B\}$$ $$=\{\emptyset,\underbrace{\{\theta_{1},\theta_{2},\theta_{3}\}}_{A},\underbrace{\{\theta_{2},\theta_{4}\}}_{B},\underbrace{\{\{\theta_{1},\theta_{2},\theta_{3}\},\{\theta_{2},\theta_{4}\}\}}_{A\cup B}\}$$ $$=\{\emptyset,\theta_{1}\cup\theta_{2}\cup\theta_{3},\theta_{2}\cup\theta_{4},\theta_{1}\cup\theta_{2}\cup\theta_{3}\cup\theta_{4}\}$$
4
If X and Y are expressed more precisely in function of the finer granules θi by example as X , {θ1} ≡
θ1 and Y , {θ2, θ3, θ4} ≡ θ2 ∪ θ3 ∪ θ4 then:

$$2^{\Theta_{2}=\{X,Y\}}=\{\emptyset,X,Y,X\cup Y\}$$ $$=\{\emptyset,\underbrace{\{\theta_{1}\}}_{X},\underbrace{\{\theta_{2},\theta_{3},\theta_{4}\}}_{Y},\underbrace{\{\{\theta_{1}\},\{\theta_{2},\theta_{3},\theta_{4}\}\}}_{X\cup Y}\}$$ $$=\{\emptyset,\theta_{1},\theta_{2}\cup\theta_{3}\cup\theta_{4},\theta_{1}\cup\theta_{2}\cup\theta_{3}\cup\theta_{4}\}$$
$$X,Y\}\,\neq2^{\Theta^{r e f}=\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\}}$$

We see that one has naturally:

$$X\cup Y=\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\}=\Theta^{r e f},$$

2 Θ1={A,B}6= 2Θ2={X,Y }6= 2Θref ={θ1,θ2,θ3,θ4}
even if working from θi with A ∪ B = X ∪ Y = {θ1, θ2, θ3, θ4} = Θref.

## - The Hyper-Power Set: Dθ , (Θ,∪,∩)

One of the cornerstones of DSmT is the free Dedekind's lattice [4] denoted *hyper-power set* in DSmT
framework. Let Θ = {θ1**, . . . , θ**n} be a finite set (called frame) of n exhaustive elements. The hyper-power set DΘ is defined as the set of all composite propositions/subsets built from elements of Θ with ∪ and ∩ operators such that:
1. ∅, θ1**, . . . , θ**n ∈ DΘ.

2. If **A, B** ∈ DΘ, then A ∩ B ∈ DΘ and A ∪ B ∈ DΘ.

3. No other elements belong to DΘ, except those obtained by using rules 1 or 2.