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If I want to say that a problem can be solved by means of introducing some constraints, do I use a definite or an indefinite article for the word introduction ?

(a) The problem can be solved through the introduction of additional constraints.

(b) The problem can be solved through an introduction of additional constraints.

To me it doesn't look like a possessive form which would require a definite article, however, the sounds more familiar to me with the word introduction. I am not sure though.

I have the same doubts about a phrase removal:

(a) This entails the removal of element 47 from X.

(b) This entails a removal of element 47 from X.

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    On both examples, I'd use the. While neither is technically incorrect, the makes it sound stronger, as though you are introducing additional constraints, rather than just that somebody should. In English, usually go with your gut.
    – Will
    Commented Jun 23, 2015 at 20:31
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    On all these I'd use by + gerund: by introducing additional constraints, by removing an element. Go with the verbiest form that works. Commented Jun 23, 2015 at 22:57
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    The short answer is you use "the".
    – Dog Lover
    Commented Jun 26, 2015 at 0:47

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The answer to your question is based on whether you're talking about "any solution" or "a definitive solution".

As stated by Glorfindel, the articles here aren't necessary and may be omitted, so the decision is whether you wish to leave room for other solutions (yours is one solution among many) which would be best served by using an, or not (yours is the proper solution) which would be best served by using the.

This may also be dependent upon the tone of the document in which this phrase will be used. If you're writing a persuasive argument, use the definite article. Otherwise, the choice is up to your personal style.

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