Logic translation involving the existential quantifier and "such that"

Question

A: "There exists an integer greater than 5 such that it is less than 10"

B: "There exists an integer such that it is greater than 5 and less than 10."

C: "There exists an integer less than 10 such that it is greater than 5."

D: "There exists an integer such that it is less than 10 and greater than 5."

I know that A can be translated to B (likewise C to D).

B and D are obviously equivalent, whereas I don't think A and C are.

Thus, I believe that A implies B but not vice-versa (likewise C implies D but not vice-versa). But is this correct??

Answer 1

They're the same. "Such that" would be the colon in my interpretation of A: $$\exists n>8 : (n<15)$$

However, in this case "n>8" is itself a statement, so what we're really saying is that there exists some n that satisfies both the conditions: $$\exists n : (n>8) \wedge (n<15)$$

Because "and" is commutative, and "such that" simplifies to "and" in this case, all of the statements are equivalent.

Before you ask, "for all" quantifier is different. If you say the following: $$\forall n>8 : n<15$$ what you're really saying is that any $n$ greater than 8 implies that $n$ is less than 15, or: $$\forall n :(n>8) \rightarrow (n<15)$$

This is the reason why the statements

for all x such that x>0, x<10

and

for all x such that x<10, x>0

are not the same. However, the statement:

there exists an x greater than zero such that x is less than ten

is equivalent to

there exists an x less than ten such that x is greater than zero

There are lots of linguistic issues like this. "Such that" and "but" are the ones I've had the most trouble with.