“I don't have any plans for Thursday. You?”
You and your girlfriend just found out that a band you like is coming to your city next Thursday. You want to go, so you send her a text message to see if she's free that day.
I don't have any plans for Thursday. You?
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After you answer someone's question, you can ask them the same question by simply adding "You?" to the end of your answer. For example:
A: Where you from?
B: I'm from Kyoto. You?
You can also ask someone "You?" after a statement that you make:
I'm from Boston. You?
Asking "You?" is rather casual. Some other similar phrases that are less casual are:
What about you?
(someone) has plans for (a time)
"Having plans" means that you've made an appointment or a decision to do something at a certain time. You can ask someone whether they "have plans":
Do you have any plans for Saturday?
Do you have plans for later this afternoon?
You can also refuse someone's invitation by saying that you "have plans":
A: Why don't we go grab a coffee?
B: Sorry, but I actually have plans.