A Complete search space contains all solutions, while an incomplete search space only contains some possible solutions
Exhaustive search explores entire space (assuming it is finite) and will find solution if one exists
Pruning
Pruning rules out certain operators in certain states. For example, we may remove invalid moves when solving chess because there is no point in searching invalid space.
The space is still complete if no solutions are pruned
The strategy is still complete if not all solutions are pruned