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I want to write a sentence like

We consider the equation f(x)=0. We show that (the) existence of (a) solution and (the) uniqueness of (the) solution.

I want to know if the above sentence uses correctly "the" and "a". In particular, I don't know if I should put "the" in front of "existence" or "uniqueness".

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In your given sentence,....

We consider the equation f(x)=0. We show that (the) existence of (a) solution and (the) uniqueness of (the) solution.

...the word 'that' seems to make the sentence uncoordinated (assuming We show that (the) existence of (a) solution and (the) uniqueness of (the) solution isn't just part of a bigger sentence). Without the word 'that' and shortening the second clause of the second sentence, it becomes:

We consider the equation f(x)=0. We show the existence of a solution and it's uniqueness.

....which I believe is more readable. I've maintained the use of the word we in case there's a scientific style of writing you're aiming for.

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Without context, it's difficult to recommend the best way to write these sentences..

Here is one approach -

Consider the equation f(x)=0. This shows both (the) existence of (a) solution and (the) uniqueness of (this) solution.

My reasoning: I don't see any added value to using 'we.' It's confusing in this usage and distracts us from the main point.

The sentences are focused on the equation, and that's how I've written everything - focusing on that.

Then, we're pointing out how the solution is other key part of what's being discussed. So, everything else is done to support this.

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