## [Part1] Consistency

##### Question 1: Consistency AllDifferent

We consider the $$allDifferent(x_1,x_2,x_3,x_4)$$ constraint with domains $$D(x_1)=\{1,3,4\}, D(x_2)=\{1,3\}, D(x_3)=\{1,3\}, D(x_4)=\{1,4,5\}$$.

Select all the correct statements.

##### Question 2:

A binary constraint $$C(X,Y)$$ is represented visually (feasible solutions inside the green shapes).

The domains $$D(X)$$ and $$D(Y)$$ are also represented as small green rectangles on the axis.

Select all the true statements given the current represented domains.

##### Question 3: General assertions

Select all the correct statements given a constraint $$C(X,Y,Z)$$ with three finate integer domain variable in its scope.

##### Question 4: Range consistency

C is range-consistent iff

$$\forall 1 \leq i \leq n, \forall v_i \in D(i):\quad \exists (v_1,\ldots,v_{i-1},v_{i+1},\ldots,v_n) \in$$ $$[\min(D_1),\max(D_1)] \times \ldots \times [\min(D_{i-1}),\max(D_{i-1})] \times [\min(D_{i+1}),\max(D_{i+1})] \times \ldots \times [\min(D_n),\max(D_n)]$$ $$\text{ such that } C(v_1,\ldots,v_i,\ldots,v_n)$$

Select all the correct statements

##### Question 5: Binary Sum

Consider the constraint $$X+Y=0$$ with $$D(X)=[100,110]$$ and $$D(Y)=[-1000,120] \cap [-105, 1000]$$. Assuming a consistent (does not remove any solution) filtering algorithm for the constraint that performs a bound-consistency (after its filtering the constraint is bound-consistent), what will be $$D(X)$$?

Write your answer under the format $$\{v_1,v_2,...,v_n\}$$ with $$v_i < v_i+1$$. For instance $$\{6,10,13\}$$ but not $$\{13,6,10\}$$.