I'd say "a" in both cases.
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BelieverHi,"a" because we are not talking about a particular carpenter or saw, so we can't use "the".
Please tell me which articles would you use for the underlined parts and possibly tell me why.
Carpenters are people who make things with wood. One tool used by the (or 'a' ???) carpenter is a (the ???) saw.
J LewisWe can ask Who invented saws? or Who invented the saw?I think here only "the" is possible.
J LewisThe definite article is also possible in both casesOf course it is, and I am sure this has been mentioned to Believer before. He may want to read the full page at this link, so we get less questions on the article:-)