Although, keeping trimmed hair or shaved dog might cause less allergic symptoms compare to hairy one. We have a cat and until a month ago allergies were getting high in our home because she was hairy and shedding hair alot. We decided to get her shaved and now things have calmed down.
Air purfiers are a great idea esp if the allergy is a mild one. But even with its use, most pets still have "dander" which is the allergen that causes all those annoying symptoms.
A "hypo-allergenic" pet is one that does not have this dander. Their hair is more similar to human hair and does not produce allergic symptoms. Poodles are well known for this trait. Other dogs have been bred to have their type of hypo-allergenic hair....same with cats. Cats are more difficult unless you get one of those horriffically ugly "hairless" breeds (the kind that Seinfeld complained about, shouting "OMG!! Who turned that cat inside out?!!!" lol Anyway, there are a couple of newly bred hypo-allergenic billies but they are so super expensive that most of us would not be able to afford...
Some people have allergies that will cause discomfort even if they have an air purifier in every room. So a "hypo-allergenic" pet will allow those types to have pets without the suffering.