1. May I help you to some more vegetables?

2. May I give you some more vegetables?
(Are both the same in meaning? Do you use #1 frequently?)

3. Help yourself to the cake.
4. Give yourself the cake.
(What about this pair? Does #4 mean #3?)

1 and 2 mean the same thing, yes.

I, myself, rarely use either expression. I would say, "Would you like more vegetables?"

3 is a common expression, yes, but 4 is not. I would say, "Have some cake!" or "Would you like some cake?"

I might also say, "Would you like a piece of cake?" To say, "Help yourself to the cake" might imply the whole cake. :)

1. "May I help you to some more vegetables?" and "May I give you some more vegetables?" have a similar meaning, which is offering someone additional vegetables. Both phrases imply that the person offering is willing to assist in providing more vegetables. However, the use of these phrases may vary depending on the context and personal preference.

2. Although both phrases have a similar meaning, the usage may differ. "May I give you some more vegetables?" is a more direct way of offering someone additional vegetables, while "May I help you to some more vegetables?" could be seen as a slightly more polite and indirect way of making the offer.

3. "Help yourself to the cake" and "Give yourself the cake" do not have the exact same meaning.

3a. "Help yourself to the cake" means that you are allowing or encouraging someone to serve themselves some cake without further assistance. It suggests that the person is free to take as much cake as they want or feel comfortable with.

4a. On the other hand, "Give yourself the cake" implies that the person speaking is giving permission for someone to take possession of the cake or to take it for themselves. It doesn't necessarily convey the idea of taking the cake without assistance or limits, as it focuses more on granting permission.

In summary, while these phrases have similar intentions, the specific meanings can vary. "Help yourself to the cake" encourages someone to take as much cake as they desire, whereas "Give yourself the cake" grants permission for someone to take possession of the cake.