All the full nodes in Bitcoin network validate a new block as soon as it is mined and propagated on the network, by verifying the nonce and the resulting block hash. Now the question is that, Do full nodes validate each transaction in the Block, based on its complete history of input and outputs to check for any error?

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Yes, all transactions in every block are verified by full nodes. Nodes do not have to traverse the full history though, because they would have already validated the parent transactions before the child block is received usually. So they can just make sure the new transaction is valid in the ways it needs to be.

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