What exactly is the difference between "introduction to" and "introduction of"?
For example: should it be "Introduction to the problem" or "Introduction of the problem"?
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There are two different meanings here:
(1) the action of introducing something
issues arising from the introduction of new technology
(2) a thing preliminary to something else, especially an explanatory section at the beginning of a book, report, or speech
a good general introduction to the subject is A Social History of England
I suspect you want meaning (2):
The first paragraph provides an introduction to the problem.
It's a matter of meaning and usage.
Both of "introduction of" and "introduction to" are grammatical and have the same meaning in the following case:
Do you mind giving us an introduction of the problem?
Do you mind giving us an introduction to the problem?
But, they will have different meanings in the case below:
Your introduction to the problem will last 5 minutes. ("You" are going to be introduced)
Your introduction of the problem will last 5 minutes. (The meaning here is ambiguous: 1) "You" are going to be introduced or 2) "You" are the one who introduced the problem to others)
It's a matter of usage and meaning, some examples:
"Our students are learning English faster since the introduction of new classroom methods."
"With the introduction of penicillin, pneumonia was no longer as life-threatening as before."
"Can you give me an introduction to the president of the company?"
"Since my introduction to the use of the DVD player, I watch more movies than before."
So, for your case, "introduction of the problem" should be correct, according to first one.