Arc consistency and forward checking

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Arc consistency and forward checking

I am a little bit confused from the examples in the lecture regarding forward checking and arc consistency.

Example 10.4.1 (forward checking):

  • In this example in order to satisfy the constraint we only look for at least one variable that satisfies the constraint (in V3 that is the number 3).
  • However we see that the number 3 from V3 does not satisfy the constraint with the number 3 in V2.
  • So we can say that only at least one variable should satisfy the constrain.

Example 10.4.11 (arc consistency):

  • Here the constraints are satisfied, if every number from each variable satisfies the constraints.

My question:
-Why in the first example we look for at least one number (in a variable) and in the second example we look for every single number (in a variable) that satisfies the constraint?

I know that those are two different algorithms, however i am confused about the fact, that we try to solve the “same problems” with different meaning of the constraints…