How to say "You are able to" or "You can do this"

Question

Title says it all. I am looking for sentence structures that can convey this meaning.

Answer 1

To say "you can/may ____" or "it's okay to ____", use this structure:

[verb, te-form] + も (optional) + いい

This can also be used with negatives to say "it's okay not to ____"

Positive Example: 「入っていいですよ」 (inviting someone to come into your house/room)

Positive Example (casual): 「入っていいよ」

Negative Example: 「心配しなくていいですよ」 "It's okay"/"Don't worry (about it)"; literally "it's okay not to worry"

Negative Example (casual): 「心配しなくていいよ」

Notes:

• も, like a lot of other particles, is usually ommitted when spoken or in chat and included in formal, written communication like e-mail, letters, papers, books, etc.
• The sentence ending ~よ is often tagged on the end of sentences that use this structure.

Answer 2

It differs depending on the connotation you want to attach to 'able to'

1. Ability : できる or the 可能形 (ability form of verbs)
   For example, 自分で料理できますか？means are you able to do the cooking yourself? ピアノが弾けます (possibility form of 弾く) means I am able to play the piano.

2. Permission : でもいい？ or 〜っていいよ
   ちょっとトイレに行ってもいい？ Means can I go to the toilet for a bit? This indicates not the ability to go to the toilet but rather the permission to go to the toilet.

3. Possibility : 可能
   Literally the word means possibility, and this refers more to things which are physically possible, or possible within a set of rules. Also is a more formal form of no.1.
   For example, 一年で日本語を学ぶことは可能ですか？ means is it possible to learn Japanese in a year.