# How to write the proof for $\forall x \in D \, (A(x) \to B(x))$ in standard English.

Question: How to write the proof for $$\forall x \in D \, (A(x) \to B(x))$$ in standard English.

I recently started reading a book that begins with a short introduction to induction. Although I have a lot of experience writing proofs using induction, I mainly write these proofs in another language. Therefore, I am slightly confused with the specific word choice in English. When trying to phrase for a proof that shows $$\forall n \geq a(n \in S \to n + 1 \in S)$$, I wrote the following:

Suppose n belongs to S for all $$n \geq a$$,

... ...

then n+1 also belongs to S.

I'm afraid the beginning of the proof sounds like a complete statement stating that $$\forall n \geq a(n \in S)$$, which is clearly not what I intended.

I have thought about some alternatives: the book states that wherever some integer $$n \geq a$$ belongs to S, then n+1 also belongs to S. However, I am unsure about the usage of "some" as it often implies existence. Alternatively, I was thinking about writing suppose n belongs to S where $$n \geq a$$, then we want to show n+1 is also in S, and the proof continues. However, I felt that this doesn't clearly identify that the statement holds for all n. Additionally, I'm less inclined to write the phrase "we want to show".

So what word choice should I adopt to write this proof? In general, how should I translate the above formula written with logic symbols to standard English?

• “Suppose that for every $n\ge a$, if $n\in S$ then $n+1 \in S$.“ More generally, for the schema in the title of your question, “Every $D$ that’s an $A$ is (also) a $B$.” Jul 29, 2022 at 22:33

$$\forall x\in D~(x\in A\to x\in B)$$ says "For every thing in D, if that thing in A , then that thing is in B."

More briefly, "Everything in D that is in A will be in B"

Take any thing in D. Suppose it is in A. ...(insert a valid derivation).... So such a thing will be in B.

• Thank you for your answer. However, I don't think it's possible to write "For everything in D, if that thing in A", then a bunch of words, and "Therefore that thing is in B" in a proof. Jul 27, 2022 at 2:31
• .... Why not ? @wsz_fantasy Jul 27, 2022 at 2:37
• I thought people normally use the word "suppose" instead "if" if they want to keep the assumption for many sentences as explained here(math.stackexchange.com/questions/4028574/suppose-versus-if). Jul 27, 2022 at 2:57

$$\forall n{\geq}a\;(n \in S \to n + 1 \in S)\tag1$$

• Suppose $$n$$ belongs to $$S$$ where $$n \geq a$$, then we want to show $$n+1$$ is also in $$S.$$

This is pretty okay (though I'd replace “where” with “and”), as it suggests the equivalent statement $$\forall n\;(n \in S\;\text{and}\;n\geq a\;\implies\;n + 1 \in S).\tag2$$ In fact, noting that all three statements $$\forall n{\in}S\;(n\geq a\;\implies\;n + 1 \in S)\tag3$$ are equivalent to one another, here are some alternatives (and more at ‘Any’ versus ‘arbitrary’):

• Let $$n$$ be arbitrary and suppose that $$n\in S$$ and that $$n\ge a\ldots$$
• Let $$n$$ be an arbitrary element of $$S,$$ and suppose that $$n\ge a\ldots$$
• Let $$n$$ be an arbitrary element of $$S$$ such that $$n\ge a\ldots$$

However, I feel that this doesn't clearly identify that the statement holds for all $$n.$$

Yes, because at this point you are referring to a representative value of $$n.$$ Then, when wrapping up the proof, you conclude generally that “for each $$n\ldots$$”, by relying on Universal Introduction.

Additionally, I'm less inclined to write the phrase “we want to show”.

I think the phrase is reader-friendly!

@ryan I think the main part I was missing is the “arbitrary element”.

The $$n$$ in your suggestion is tacitly understood as arbitrary.

user21820:

Contrary to ryang's post, I recommend avoiding "let" for ∀-quantification.

This really is a non-objection: it is the word “arbitrary” (tacit or otherwise)—not the word “let”— that signals ∀.

Suppose $$n\geq a$$ is such that $$n\in S$$. Then ⟨your proof here⟩, and thus $$n+1\in S$$.
Varying $$n$$ in the above argument, we see that, for any $$n\geq a$$, the implication $${n\in S}\to{n+1\in S}$$ holds.
Contrary to ryang's post, I recommend avoiding "let" for $$∀$$-quantification. If you want actual idiomatic English, you should not use "let". The idiomatic alternative is: