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How Exchanges (Binance, Coinbase) Record User Trade Into Blockchain

My question is how cryptocurrency exchanges like Binance or Coinbase record users trade into blockchain technically. For example when Mr.X buys BTC/USDT pair from Mr.Y, how does the exchange record these two Transactions into the blockchain. as far as I know, We have two transactions now, Mr.X has to sign a USDT transaction to Mr. Y's account & Mr.Y has to sign a transaction for Mr. X's BTC wallet on that exchange. I was wondering how exchanges manage network fees? & how they record these trades into blockchain I also know Mr.X order will probably fulfill with different accounts but I just want to simplify my question thank you.

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Answer

As far as I know both of them are centralized entities so what happens is:

Mr X and Mr Y send their crypto to the exchange controlled wallets. The exchanges keep the record in their own DB/ledger who owns what. Then when Mr Y and Mr X make a trade the exchanges record that trade in their own DB/ledger and nothing is recorded on public blockchain. Now Mr X withdraws the crypto from one of the exchanges and it gets recorded on public blockchain (If he withdraws BTC then it will be recorded on BTC blockchain). So Mr X and Mr Y can do thousands of trades without paying network fees as everything happens in the exchange's internal system (they may have per trade fee)

DeFi - I have no idea how it works, as didn't dig into it :)

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source: stackoverflow.com
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