I'm having a hard time understanding the logic behind my lecturer's predicate statement:
$∀x ∈ \{1,2,3\}\; ∃y ∈ \{4,5,6,7\} y|x$
We have to find the truth value from the above statement. Ignore the fact that I used 1,2,3 etc. but please read my reasoning below and tell me the best way to go about solving this.
I am assuming that the | means division in this scenario. When I first saw this statement, here is my thinking and the problem that I encountered:
I did a table that calculates y divided by x and got a mix of integers see working out on paper
The predicate statement, do however, state that: some of the y values divides all of the x values or all of the x values are divisible by some of the y values But I am not sure what to look for here.
What does finding the truth value from this statement even mean? All I yielded were some integers from the calculation.
The order of the quantifiers is important here.
$\forall x \in [1,2,3] \exists y \in [4,5,6,7] y|x$ is read as "for every $x$, I can pick a $y$ such that $x$ is divisible by $y$." So we pick the $y$ after we choose $x.$
However, $\exists y \in [4,5,6,7] \forall x \in [1,2,3] y|x$ is read as "there is a $y$ that I can pick such that for every $x$ you could pick, $x$ is divisible by $y$." So the $x$ is chosen after the $y.$