Logic statements

Question

Mark with T or F all the below statements in such a way that they do not contradict with each other:

1. At most $1$ statement is true

2. At most $1$ statement is false

3. At most $2$ statements are true

4. At most $2$ statements are false

5. At most $3$ statements are true

6. At most $3$ statements are false

7. At most $4$ statements are true

8. At most $4$ statements are false

9. At most $5$ statements are true

10. At most $5$ statements are false

We have a total of $10$ statements. If we mark the $1^{\mathrm{st}}$ with T, it means all others must be false. But this contradicts with the 2nd. We can safely mark the last $2$ as True. But if we say, for example, at most $5$ statements are true, can it be also correct that "at most $3$ (or $2$ or $4$, etc.) statements are true"? At most $5$ statements are true means at least $5$ statements are false, which is OK in combination with the $10^{\mathrm{th}}$, right?

But if we say "at most $1$ statement is true", it means "at minimum $9$ statements are false". But if "at minimum $9$ statements are false", doesn't it mean also that "at minimum $7$ statements are false" for example?

How can we combine them all together?

I am completely stuck!

Answer 1

If we mark the 1st with T, it means all others must be false. But this contradicts with the 2nd. We can safely mark the last 2 as True.

I don't understand your reasoning here. Why are you sure you can safely mark 2 as "true?" You've just argued that 2 is false if 1 is true, so you need to be sure that 1 is false ...

(Now, 1 is in fact guaranteed to be false, but you haven't argued that yet.)

if we say, for example, at most 5 statements are true, can it be also correct that "at most 3 statements are true"?

Sure - maybe exactly 3 statements are true.

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As a hint to get started, based on the two comments above:

Can you show that if the statement "at most $n$ statements are true" (resp. false) is true, then the statement "at most $k$ statements are true" (resp. false) is also true if $k>n$?

E.g. if 1 is true, then 3 has to be true as well, etc. This lets you rule out a bunch of statements right off the bat.

Another hint:

Try to solve a simpler case - say, what if you had only statements 1-4? What about 1-6?