It is forbidden on Shabbat to take items from a private domain (like one's house) to a public domain (like the street). For example, if the owner of a house stands inside, takes an object, lifts it up, transfers it outside, and puts it in the basket of a poor person; or if the poor person takes the object from the owner of the house, lifts it up, and puts it in his basket.
The Sages added two more prohibitions, where the action is done jointly. For example, if the owner of the house takes the object, lifts it up, and the poor person accepts it while outside. Neither of them has completely violated Shabbat.
There are also four cases of bringing an object in. This makes a total of eight, and is not parallel to the “two that are four!” To answer this, we count only those cases where one is liable for a Torah violation, and equate taking out to bringing in, because they are both in essence changing the domain - and you will get exactly two that are four.
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