False, as you've described it here, because private and public keys work a little differently than that. The underlying math is tricky, but I'll try and give a common sense explanation in everyday language instead.
Imagine that there was a special metal called Cryptonium, harder than anything else on earth and completely indestructible. No one else knows it exists, much less how to make things out of it. Let's also say you had a special machine which could make plaques out of Cryptonium, with raised letters on them saying whatever you wanted. Obviously, you would keep the existence of this plaque-making machine a closely guarded secret. You probably wouldn't even want anyone to know it was you that owned it. But you still want to share this exciting discovery with the world, so you secretly produce a message that says "Behold, a new and indestructible metal!" Then, late at night, you sneak into the middle of town and put the Cryptonium plaque up in the town square.
The next day, everyone gathers in the town square to wonder at this amazing new plaque. They take turns bashing at it with the hardest, sharpest, heaviest things they can find, and the plaque remains perfectly unscratched. "Sure enough," they figure, "it's indestructible."
You decide to use the secret machine to write more things, helping out the townspeople and gradually earning a reputation as a trustworthy and reliable source, even though nobody knows who you are. But a few months later something terrible happens. New plaques start showing up in town during the night, and they are fakes!! Someone else is taking advantage of your reputation to spread false and misleading messages! They don't have your special machine of course--they're making their plaques out of platinum, and it's only almost indestructible. But the two metals look the same, and the townspeople don't know the difference. What will you do?
You could announce yourself to the townspeople, show them the machine, and tell them the new plaques are fakes. They might believe you, but your machine wouldn't be a secret anymore. So instead you do something really clever. You use the machine to make a small, square plaque in the shape of a hammer's head. On the plaque, you put the message "This hammer will destroy all fake plaques." Then, you attach a handle to the hammer head you've made and leave it in the middle of town the next night.
Now, the townspeople can simply use the hammer to break all the fake platinum plaques. Sure enough, the false and misleading messages are all easily destroyed, while the ones you've produced are able to withstand the hammer's force. Your reputation is preserved, and your secret was never revealed. Together the hammer, the secret machine, and the plaques form an unbeatable system that lets you send messages which could only have come from you--without revealing who you really are. Not bad, hey?
In many ways public key cryptography works like our imaginary Cryptonium metal. It's made up of three parts, and it gives other people a way to be certain a message came from you without you having to reveal your identity or your secret way of making messages. Like the Cryptonium system, you need all three components to make the system work. These three parts are your public key, your private key, and your signatures (which you can make as many of as you like, signing all different messages.) In our story, the private key is like the secret machine, the public key is like the hammer, and the signatures are like the plaques. In reality public key cryptography is more complex than I've described here, and if you want to learn more about it you can check out this course from Khan Academy. However this example should be good enough to understand how private and public keys are used in Bitcoin.
First, nobody else ever sees your private key. Instead, like the secret machine, you use your private key to make signatures (like the plaques) that people can be certain came from you. Each person (and each bitcoin address) has both a public and a private key, which share a special mathematical relationship, and the public key (like the hammer) is what other people can use to verify your signatures are real. When you send a Bitcoin transaction, you don't give away your secret machine--you only use it to make a message saying who you're giving the bitcoins to. This signed message, or signature, couldn't have come from anyone else except the owner of the address, so then everyone knows the transaction is real.
Second, private keys don't come from miners--they come from you. You're the one who secretly chooses a random private key (the Bitcoin software does this for you), and that private key was used to figure out your address (which is basically a version of your public key) before anyone ever sent it some bitcoins. Miners never have it to begin with. They only see your transactions (signatures) and your receiving address (public key).
Finally, just to confuse things a little bit, you have to use up a whole transaction at one time. So if someone sent you 10 bitcoins, and you want to send someone else 5 of them, you actually send out a transaction that takes all 10 that you received, sends 5 to the other person's address, and 5 to another address of your own (this is called the "change" from your transaction). Bitcoin does this so that other people can't tell which part of the transaction is what you sent and which part is the change (nobody knows which addresses belong to the same person). It helps protect your privacy. But if you wanted to, you could send the change back to the same address, because your private keys are still safe. You'd just have to use a different client than the default Bitcoin client, because it doesn't have a way to do that. All of this happens behind the scenes, which is why your balance still shows as 5 bitcoins after you do this. Bitcoin stores all your different addresses and private keys in something called a wallet, and the balance is the total bitcoins owned by the wallet. So it keeps everything simple up front!
Make sense? Hope it wasn't too long, but I wanted to try and make the public/private key thing more clear.
