The verification process (where trust is important) of transactions has to prevent two things:
You can only send what you own, and you can only send it once!
In real world it is simple. If you spend a 10$ bill then it is gone and you cannot spend it a second time. As in DLT everything is electronic this is more difficult to track and the main challenge, prevent double spend, has to be solved in an appropriate way.
Bitcoin uses Pow (Proof of work) to enable trust and verify transactions. In PoW a miner (or node) solves a highly complex mathematical cryptographic problem. The first miner to solve the problem is allowed to create the next block and put the pending transactions into that block (see previous chapter). This process is called mining as it requires a lot of energy and power, similar to mining tunnels into a mountain.
XRP uses the consensus mechanism. Simplified it requires a majority of distributed validators to agree on the sequence of transactions. Thus when from the same account two transactions are initiated, one will be the first. The second transaction then only can be validated if the balance on the account is still high enough.
All DLT systems write the verified transactions on every node/validator onto the new ledger, or ledger state.
At this point we can trust! If one of the nodes would change a transaction the appropriate block would not be identical anymore to the block the other nodes keep in their chain.
And this is the stroke of genius of this technology: One does not need to trust a single entity anymore, eg. your bank, but one trusts the technology as such.
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